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Theorem inf3lema 8062
 Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8073 for detailed description. (Contributed by NM, 28-Oct-1996.)
Hypotheses
Ref Expression
inf3lem.1
inf3lem.2
inf3lem.3
inf3lem.4
Assertion
Ref Expression
inf3lema
Distinct variable group:   ,,

Proof of Theorem inf3lema
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ineq1 3692 . . 3
21sseq1d 3530 . 2
3 inf3lem.4 . . 3
4 sseq2 3525 . . . . 5
54rabbidv 3101 . . . 4
6 inf3lem.1 . . . . 5
7 sseq2 3525 . . . . . . . 8
87rabbidv 3101 . . . . . . 7
9 ineq1 3692 . . . . . . . . 9
109sseq1d 3530 . . . . . . . 8
1110cbvrabv 3108 . . . . . . 7
128, 11syl6eq 2514 . . . . . 6
1312cbvmptv 4543 . . . . 5
146, 13eqtri 2486 . . . 4
15 vex 3112 . . . . 5
1615rabex 4603 . . . 4
175, 14, 16fvmpt 5956 . . 3
183, 17ax-mp 5 . 2
192, 18elrab2 3259 1
 Colors of variables: wff setvar class Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  {crab 2811   cvv 3109  i^icin 3474  C_wss 3475   c0 3784  e.cmpt 4510  |cres 5006  cfv 5593   com 6700  reccrdg 7094 This theorem is referenced by:  inf3lemd  8065  inf3lem1  8066  inf3lem2  8067  inf3lem3  8068 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601
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