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Theorem inf3lema 7571
 Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7582 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:   ,,
Allowed substitution hints:   (,,)   (,,)   (,,)   (,,)

Proof of Theorem inf3lema
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ineq1 3527 . . 3
21sseq1d 3367 . 2
3 inf3lem.4 . . 3
4 sseq2 3362 . . . . 5
54rabbidv 2940 . . . 4
6 inf3lem.1 . . . . 5
7 sseq2 3362 . . . . . . . 8
87rabbidv 2940 . . . . . . 7
9 ineq1 3527 . . . . . . . . 9
109sseq1d 3367 . . . . . . . 8
1110cbvrabv 2947 . . . . . . 7
128, 11syl6eq 2483 . . . . . 6
1312cbvmptv 4292 . . . . 5
146, 13eqtri 2455 . . . 4
15 vex 2951 . . . . 5
1615rabex 4346 . . . 4
175, 14, 16fvmpt 5798 . . 3
183, 17ax-mp 8 . 2
192, 18elrab2 3086 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725  crab 2701  cvv 2948   cin 3311   wss 3312  c0 3620   cmpt 4258  com 4837   cres 4872  cfv 5446  crdg 6659 This theorem is referenced by:  inf3lemd  7574  inf3lem1  7575  inf3lem2  7576  inf3lem3  7577 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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454
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