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Unicode version
Theorem infeq5 4593
Description: The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (shown on the right-hand side in the form of omex 4599.) The left-hand side provides us with a very short way to express of the Axiom of Infinity using only elementary symbols. This proof of equivalence does not depend on the Axiom of Infinity.
Assertion
Ref Expression
infeq5 |- (E.x x (. U.x <-> om e. V)
Proof of Theorem infeq5
StepHypRef Expression
1 df-pss 2045 . . . . 5 |- (x (. U.x <-> (x (_ U.x /\ x =/= U.x))
2 unieq 2500 . . . . . . . . . 10 |- (x = (/) -> U.x = U.(/))
3 uni0 2515 . . . . . . . . . 10 |- U.(/) = (/)
42, 3syl6req 1516 . . . . . . . . 9 |- (x = (/) -> (/) = U.x)
5 eqtrt 1484 . . . . . . . . 9 |- ((x = (/) /\ (/) = U.x) -> x = U.x)
64, 5mpdan 702 . . . . . . . 8 |- (x = (/) -> x = U.x)
76necon3i 1597 . . . . . . 7 |- (x =/= U.x -> x =/= (/))
87anim1i 334 . . . . . 6 |- ((x =/= U.x /\ x (_ U.x) -> (x =/= (/) /\ x (_ U.x))
98ancoms 436 . . . . 5 |- ((x (_ U.x /\ x =/= U.x) -> (x =/= (/) /\ x (_ U.x))
101, 9sylbi 199 . . . 4 |- (x (. U.x -> (x =/= (/) /\ x (_ U.x))
111019.22i 1036 . . 3 |- (E.x x (. U.x -> E.x(x =/= (/) /\ x (_ U.x))
12 eqid 1468 . . . . 5 |- {<.y, z>. | z = {w e. x | (w i^i x) (_ y}} = {<.y, z>. | z = {w e. x | (w i^i x) (_ y}}
13 eqid 1468 . . . . 5 |- (rec({<.y, z>. | z = {w e. x | (w i^i x) (_ y}}, (/)) |` om) = (rec({<.y, z>. | z = {w e. x | (w i^i x) (_ y}}, (/)) |` om)
14 visset 1804 . . . . 5 |- x e. V
1512, 13, 14, 14inf3lem7 4591 . . . 4 |- ((x =/= (/) /\ x (_ U.x) -> om e. V)
161519.23aiv 1290 . . 3 |- (E.x(x =/= (/) /\ x (_ U.x) -> om e. V)
1711, 16syl 10 . 2 |- (E.x x (. U.x -> om e. V)
18 difexg 2712 . . 3 |- (om e. V -> (om \ {(/)}) e. V)
19 0ex 2701 . . . . . . 7 |- (/) e. V
2019snid 2425 . . . . . 6 |- (/) e. {(/)}
21 disj4 2307 . . . . . . . . 9 |- ((om i^i {(/)}) = (/) <-> -. (om \ {(/)}) (. om)
22 disj3 2304 . . . . . . . . 9 |- ((om i^i {(/)}) = (/) <-> om = (om \ {(/)}))
2321, 22bitr3 175 . . . . . . . 8 |- (-. (om \ {(/)}) (. om <-> om = (om \ {(/)}))
24 peano1 3139 . . . . . . . . . . 11 |- (/) e. om
25 eleq2 1527 . . . . . . . . . . 11 |- (om = (om \ {(/)}) -> ((/) e. om <-> (/) e. (om \ {(/)})))
2624, 25mpbii 193 . . . . . . . . . 10 |- (om = (om \ {(/)}) -> (/) e. (om \ {(/)}))
27 eldif 2047 . . . . . . . . . 10 |- ((/) e. (om \ {(/)}) <-> ((/) e. om /\ -. (/) e. {(/)}))
2826, 27sylib 198 . . . . . . . . 9 |- (om = (om \ {(/)}) -> ((/) e. om /\ -. (/) e. {(/)}))
2928pm3.27d 325 . . . . . . . 8 |- (om = (om \ {(/)}) -> -. (/) e. {(/)})
3023, 29sylbi 199 . . . . . . 7 |- (-. (om \ {(/)}) (. om -> -. (/) e. {(/)})
3130a3i 74 . . . . . 6 |- ((/) e. {(/)} -> (om \ {(/)}) (. om)
3220, 31ax-mp 7 . . . . 5 |- (om \ {(/)}) (. om
33 unidif0 2729 . . . . . . 7 |- U.(om \ {(/)}) = U.om
34 limom 3136 . . . . . . . 8 |- Lim om
35 limuni 3019 . . . . . . . 8 |- (Lim om -> om = U.om)
3634, 35ax-mp 7 . . . . . . 7 |- om = U.om
3733, 36eqtr4 1490 . . . . . 6 |- U.(om \ {(/)}) = om
3837psseq2i 2128 . . . . 5 |- ((om \ {(/)}) (. U.(om \ {(/)}) <-> (om \ {(/)}) (. om)
3932, 38mpbir 190 . . . 4 |- (om \ {(/)}) (. U.(om \ {(/)})
40 psseq1 2125 . . . . . 6 |- (x = (om \ {(/)}) -> (x (. U.x <-> (om \ {(/)}) (. U.x))
41 unieq 2500 . . . . . . 7 |- (x = (om \ {(/)}) -> U.x = U.(om \ {(/)}))
4241psseq2d 2131 . . . . . 6 |- (x = (om \ {(/)}) -> ((om \ {(/)}) (. U.x <-> (om \ {(/)}) (. U.(om \ {(/)})))
4340, 42bitrd 526 . . . . 5 |- (x = (om \ {(/)}) -> (x (. U.x <-> (om \ {(/)}) (. U.(om \ {(/)})))
4443cla4egv 1854 . . . 4 |- ((om \ {(/)}) e. V -> ((om \ {(/)}) (. U.(om \ {(/)}) -> E.x x (. U.x))
4539, 44mpi 44 . . 3 |- ((om \ {(/)}) e. V -> E.x x (. U.x)
4618, 45syl 10 . 2 |- (om e. V -> E.x x (. U.x)
4717, 46impbi 157 1 |- (E.x x (. U.x <-> om e. V)
Colors of variables: wff set class
Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977   =/= wne 1577  {crab 1640  Vcvv 1802   \ cdif 2034   i^i cin 2036   (_ wss 2037   (. wpss 2038  (/)c0 2270  {csn 2399  U.cuni 2493  {copab 2656  Lim wlim 2939  omcom 3121   |` cres 3162  reccrdg 3916
This theorem is referenced by:  inf5 4600
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  ax-reg 4565
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-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-iun 2558  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-lim 2943  df-suc 2944  df-om 3122  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-fv 3188  df-rdg 3917
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