Theorem zfcndinf 9017

Description: Axiom of Infinity ax-inf 8076, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 4634 in the proof. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by NM, 15-Aug-2003.)

Assertion

Ref	Expression
zfcndinf

Distinct variable group:   ,,,

Proof of Theorem zfcndinf

Step	Hyp	Ref	Expression
1		el 4634	. . 3
2		nfv 1707	. . . . . 6
3		nfe1 1840	. . . . . . . 8
4	2, 3	nfim 1920	. . . . . . 7
5	4	nfal 1947	. . . . . 6
6	2, 5	nfan 1928	. . . . 5
7	6	nfex 1948	. . . 4
8		axinfnd 9005	. . . . 5
9	8	19.37iv 1769	. . . 4
10	7, 9	exlimi 1912	. . 3
11	1, 10	ax-mp 5	. 2
12		elequ1 1821	. . . . . 6
13		elequ1 1821	. . . . . . . 8
14	13	anbi1d 704	. . . . . . 7
15	14	exbidv 1714	. . . . . 6
16	12, 15	imbi12d 320	. . . . 5
17	16	cbvalv 2023	. . . 4
18	17	anbi2i 694	. . 3
19	18	exbii 1667	. 2
20	11, 19	mpbir 209	1

Syntax hints:  ->wi 4  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818

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-8 1820  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-pow 4630  ax-pr 4691  ax-reg 8039  ax-inf 8076

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-un 3480  df-nul 3785  df-sn 4030  df-pr 4032