Theorem epfrc 4870

Description: A subset of an epsilon-founded class has a minimal element. (Contributed by NM, 17-Feb-2004.) (Revised by David Abernethy, 22-Feb-2011.)

Hypothesis

Ref	Expression
epfrc.1

Assertion

Ref	Expression
epfrc

Distinct variable groups:   ,   ,

Proof of Theorem epfrc

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		epfrc.1	. . 3
2	1	frc 4850	. 2
3		dfin5 3483	. . . . 5
4		epel 4799	. . . . . . 7
5	4	a1i 11	. . . . . 6
6	5	rabbiia 3098	. . . . 5
7	3, 6	eqtr4i 2489	. . . 4
8	7	eqeq1i 2464	. . 3
9	8	rexbii 2959	. 2
10	2, 9	sylibr 212	1

Syntax hints:  ->wi 4  <->wb 184  /\w3a 973  =wceq 1395  e.wcel 1818  =/=wne 2652  E.wrex 2808  {crab 2811  cvv 3109  i^icin 3474  C_wss 3475  c0 3784   class class class wbr 4452  cep 4794  Frwfr 4840

This theorem is referenced by:  wefrc  4878  onfr  4922  epfrs  8183

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-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-br 4453  df-opab 4511  df-eprel 4796  df-fr 4843