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Theorem infeq5 7584
 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 7590.) The left-hand side provides us with a very short way to express the Axiom of Infinity using only elementary symbols. This proof of equivalence does not depend on the Axiom of Infinity. (Contributed by NM, 23-Mar-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
infeq5

Proof of Theorem infeq5
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pss 3328 . . . . 5
2 unieq 4016 . . . . . . . . . 10
3 uni0 4034 . . . . . . . . . 10
42, 3syl6req 2484 . . . . . . . . 9
5 eqtr 2452 . . . . . . . . 9
64, 5mpdan 650 . . . . . . . 8
76necon3i 2637 . . . . . . 7
87anim1i 552 . . . . . 6
98ancoms 440 . . . . 5
101, 9sylbi 188 . . . 4
1110eximi 1585 . . 3
12 eqid 2435 . . . . 5
13 eqid 2435 . . . . 5
14 vex 2951 . . . . 5
1512, 13, 14, 14inf3lem7 7581 . . . 4
1615exlimiv 1644 . . 3
1711, 16syl 16 . 2
18 infeq5i 7583 . 2
1917, 18impbii 181 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359  wex 1550   wceq 1652   wcel 1725   wne 2598  crab 2701  cvv 2948   cin 3311   wss 3312   wpss 3313  c0 3620  cuni 4007   cmpt 4258  com 4837   cres 4872  crdg 6659 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-reg 7552 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-rdg 6660
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