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.

Step	Hyp	Ref	Expression
1		ineq1 3692	. . 3
2	1	sseq1d 3530	. 2
3		inf3lem.4	. . 3
4		sseq2 3525	. . . . 5
5	4	rabbidv 3101	. . . 4
6		inf3lem.1	. . . . 5
7		sseq2 3525	. . . . . . . 8
8	7	rabbidv 3101	. . . . . . 7
9		ineq1 3692	. . . . . . . . 9
10	9	sseq1d 3530	. . . . . . . 8
11	10	cbvrabv 3108	. . . . . . 7
12	8, 11	syl6eq 2514	. . . . . 6
13	12	cbvmptv 4543	. . . . 5
14	6, 13	eqtri 2486	. . . 4
15		vex 3112	. . . . 5
16	15	rabex 4603	. . . 4
17	5, 14, 16	fvmpt 5956	. . 3
18	3, 17	ax-mp 5	. 2
19	2, 18	elrab2 3259	1

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