HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version
Theorem cdacomen 4901
Description: Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258.
Hypotheses
Ref Expression
cdacomen.1 |- A e. V
cdacomen.2 |- B e. V
Assertion
Ref Expression
cdacomen |- (A +c B) ~~ (B +c A)
Proof of Theorem cdacomen
StepHypRef Expression
1 xp01disj 4127 . . . 4 |- ((A X. {(/)}) i^i (B X. {1o})) = (/)
2 1ne0 4126 . . . . 5 |- 1o =/= (/)
3 xpsndisj 3456 . . . . 5 |- (1o =/= (/) -> ((A X. {1o}) i^i (B X. {(/)})) = (/))
42, 3ax-mp 7 . . . 4 |- ((A X. {1o}) i^i (B X. {(/)})) = (/)
5 cdacomen.1 . . . . . 6 |- A e. V
6 0ex 2701 . . . . . . 7 |- (/) e. V
75, 6xpsnen 4415 . . . . . 6 |- (A X. {(/)}) ~~ A
8 1on 4122 . . . . . . . 8 |- 1o e. On
98elisseti 1809 . . . . . . 7 |- 1o e. V
105, 9xpsnen 4415 . . . . . 6 |- (A X. {1o}) ~~ A
115, 7, 10entr4 4400 . . . . 5 |- (A X. {(/)}) ~~ (A X. {1o})
12 cdacomen.2 . . . . . 6 |- B e. V
1312, 9xpsnen 4415 . . . . . 6 |- (B X. {1o}) ~~ B
1412, 6xpsnen 4415 . . . . . 6 |- (B X. {(/)}) ~~ B
1512, 13, 14entr4 4400 . . . . 5 |- (B X. {1o}) ~~ (B X. {(/)})
16 unen 4414 . . . . 5 |- ((((A X. {(/)}) ~~ (A X. {1o}) /\ (B X. {1o}) ~~ (B X. {(/)})) /\ (((A X. {(/)}) i^i (B X. {1o})) = (/) /\ ((A X. {1o}) i^i (B X. {(/)})) = (/))) -> ((A X. {(/)}) u. (B X. {1o})) ~~ ((A X. {1o}) u. (B X. {(/)})))
1711, 15, 16mpanl12 706 . . . 4 |- ((((A X. {(/)}) i^i (B X. {1o})) = (/) /\ ((A X. {1o}) i^i (B X. {(/)})) = (/)) -> ((A X. {(/)}) u. (B X. {1o})) ~~ ((A X. {1o}) u. (B X. {(/)})))
181, 4, 17mp2an 695 . . 3 |- ((A X. {(/)}) u. (B X. {1o})) ~~ ((A X. {1o}) u. (B X. {(/)}))
19 uncom 2166 . . 3 |- ((A X. {1o}) u. (B X. {(/)})) = ((B X. {(/)}) u. (A X. {1o}))
2018, 19breqtr 2628 . 2 |- ((A X. {(/)}) u. (B X. {1o})) ~~ ((B X. {(/)}) u. (A X. {1o}))
215, 12cdaval 4892 . 2 |- (A +c B) = ((A X. {(/)}) u. (B X. {1o}))
2212, 5cdaval 4892 . 2 |- (B +c A) = ((B X. {(/)}) u. (A X. {1o}))
2320, 21, 223brtr4 2633 1 |- (A +c B) ~~ (B +c A)
Colors of variables: wff set class
Syntax hints:   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577  Vcvv 1802   u. cun 2035   i^i cin 2036  (/)c0 2270  {csn 2399   class class class wbr 2609  Oncon0 2938   X. cxp 3158  (class class class)co 3948  1oc1o 4112   ~~ cen 4348   +c ccda 4889
This theorem is referenced by:  cdadom2 4906  infcda 7510  alephadd 7524
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-v 1803  df-sbc 1932  df-csb 1992  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-int 2524  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-opr 3950  df-oprab 3951  df-1o 4117  df-er 4245  df-en 4351  df-cda 4890
Copyright terms: Public domain