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Theorem pm54.43 4546
Description: Theorem *54.43 of [WhiteheadRussell] p. 360. "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." See http://en.wikipedia.org/wiki/Principia_Mathematica#Quotations. This theorem states that two sets of cardinality 1 are disjoint iff their union has cardinality 2.

Whitehead and Russell define 1 as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 4805), so that their A e. 1 means, in our notation, A e. {x | (card` x) = 1o} i.e. (card` A) = 1o (by elab 1888) i.e. A ~~ 1o (by carden 4803 and cardnn 4796). We do not have several of their earlier lemmas available (which would otherwise be unused by our different approach to arithmetic), so our proof is longer. (It is also longer because we must show every detail.)

Theorem pm110.643 4895 shows the derivation of 1+1=2 for cardinal numbers from this theorem.

Assertion
Ref Expression
pm54.43 |- ((A ~~ 1o /\ B ~~ 1o) -> ((A i^i B) = (/) <-> (A u. B) ~~ 2o))
Proof of Theorem pm54.43
StepHypRef Expression
1 1on 4122 . . . . . . . 8 |- 1o e. On
21onirr 3087 . . . . . . 7 |- -. 1o e. 1o
3 disjsn 2431 . . . . . . 7 |- ((1o i^i {1o}) = (/) <-> -. 1o e. 1o)
42, 3mpbir 190 . . . . . 6 |- (1o i^i {1o}) = (/)
5 unen 4414 . . . . . 6 |- (((A ~~ 1o /\ B ~~ {1o}) /\ ((A i^i B) = (/) /\ (1o i^i {1o}) = (/))) -> (A u. B) ~~ (1o u. {1o}))
64, 5mpanr2 708 . . . . 5 |- (((A ~~ 1o /\ B ~~ {1o}) /\ (A i^i B) = (/)) -> (A u. B) ~~ (1o u. {1o}))
76ex 373 . . . 4 |- ((A ~~ 1o /\ B ~~ {1o}) -> ((A i^i B) = (/) -> (A u. B) ~~ (1o u. {1o})))
81elisseti 1809 . . . . . 6 |- 1o e. V
98ensn1 4405 . . . . . 6 |- {1o} ~~ 1o
108, 9ensymi 4394 . . . . 5 |- 1o ~~ {1o}
11 entrt 4395 . . . . 5 |- ((B ~~ 1o /\ 1o ~~ {1o}) -> B ~~ {1o})
1210, 11mpan2 694 . . . 4 |- (B ~~ 1o -> B ~~ {1o})
137, 12sylan2 451 . . 3 |- ((A ~~ 1o /\ B ~~ 1o) -> ((A i^i B) = (/) -> (A u. B) ~~ (1o u. {1o})))
14 df-2o 4118 . . . . 5 |- 2o = suc 1o
15 df-suc 2944 . . . . 5 |- suc 1o = (1o u. {1o})
1614, 15eqtr 1487 . . . 4 |- 2o = (1o u. {1o})
1716breq2i 2617 . . 3 |- ((A u. B) ~~ 2o <-> (A u. B) ~~ (1o u. {1o}))
1813, 17syl6ibr 213 . 2 |- ((A ~~ 1o /\ B ~~ 1o) -> ((A i^i B) = (/) -> (A u. B) ~~ 2o))
19 sneq 2407 . . . . . . . . . . . . . . 15 |- (x = y -> {x} = {y})
2019uneq2d 2174 . . . . . . . . . . . . . 14 |- (x = y -> ({x} u. {x}) = ({x} u. {y}))
21 unidm 2165 . . . . . . . . . . . . . 14 |- ({x} u. {x}) = {x}
2220, 21syl5reqr 1514 . . . . . . . . . . . . 13 |- (x = y -> ({x} u. {y}) = {x})
23 visset 1804 . . . . . . . . . . . . . . 15 |- x e. V
2423ensn1 4405 . . . . . . . . . . . . . 14 |- {x} ~~ 1o
25 1sdom2 4505 . . . . . . . . . . . . . 14 |- 1o ~< 2o
26 ensdomtr 4451 . . . . . . . . . . . . . 14 |- (({x} ~~ 1o /\ 1o ~< 2o) -> {x} ~< 2o)
2724, 25, 26mp2an 695 . . . . . . . . . . . . 13 |- {x} ~< 2o
2822, 27syl6eqbr 2642 . . . . . . . . . . . 12 |- (x = y -> ({x} u. {y}) ~< 2o)
29 sdomnen 4368 . . . . . . . . . . . 12 |- (({x} u. {y}) ~< 2o -> -. ({x} u. {y}) ~~ 2o)
3028, 29syl 10 . . . . . . . . . . 11 |- (x = y -> -. ({x} u. {y}) ~~ 2o)
3130necon2ai 1603 . . . . . . . . . 10 |- (({x} u. {y}) ~~ 2o -> x =/= y)
32 disjsn2 2432 . . . . . . . . . 10 |- (x =/= y -> ({x} i^i {y}) = (/))
3331, 32syl 10 . . . . . . . . 9 |- (({x} u. {y}) ~~ 2o -> ({x} i^i {y}) = (/))
3433a1i 8 . . . . . . . 8 |- ((A = {x} /\ B = {y}) -> (({x} u. {y}) ~~ 2o -> ({x} i^i {y}) = (/)))
35 uneq12 2169 . . . . . . . . 9 |- ((A = {x} /\ B = {y}) -> (A u. B) = ({x} u. {y}))
3635breq1d 2619 . . . . . . . 8 |- ((A = {x} /\ B = {y}) -> ((A u. B) ~~ 2o <-> ({x} u. {y}) ~~ 2o))
37 ineq12 2202 . . . . . . . . 9 |- ((A = {x} /\ B = {y}) -> (A i^i B) = ({x} i^i {y}))
3837eqeq1d 1475 . . . . . . . 8 |- ((A = {x} /\ B = {y}) -> ((A i^i B) = (/) <-> ({x} i^i {y}) = (/)))
3934, 36, 383imtr4d 541 . . . . . . 7 |- ((A = {x} /\ B = {y}) -> ((A u. B) ~~ 2o -> (A i^i B) = (/)))
4039ex 373 . . . . . 6 |- (A = {x} -> (B = {y} -> ((A u. B) ~~ 2o -> (A i^i B) = (/))))
414019.23adv 1209 . . . . 5 |- (A = {x} -> (E.y B = {y} -> ((A u. B) ~~ 2o -> (A i^i B) = (/))))
424119.23aiv 1290 . . . 4 |- (E.x A = {x} -> (E.y B = {y} -> ((A u. B) ~~ 2o -> (A i^i B) = (/))))
4342imp 350 . . 3 |- ((E.x A = {x} /\ E.y B = {y}) -> ((A u. B) ~~ 2o -> (A i^i B) = (/)))
44 en1 4407 . . 3 |- (A ~~ 1o <-> E.x A = {x})
45 en1 4407 . . 3 |- (B ~~ 1o <-> E.y B = {y})
4643, 44, 45syl2anb 455 . 2 |- ((A ~~ 1o /\ B ~~ 1o) -> ((A u. B) ~~ 2o -> (A i^i B) = (/)))
4718, 46impbid 514 1 |- ((A ~~ 1o /\ B ~~ 1o) -> ((A i^i B) = (/) <-> (A u. B) ~~ 2o))
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977   =/= wne 1577   u. cun 2035   i^i cin 2036  (/)c0 2270  {csn 2399   class class class wbr 2609  Oncon0 2938  suc csuc 2940  1oc1o 4112  2oc2o 4113   ~~ cen 4348   ~< csdm 4350
This theorem is referenced by:  pm110.643 4895
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-1o 4117  df-2o 4118  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353
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