Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  epfrc Structured version   Unicode version

Theorem epfrc 4561
 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.
StepHypRef Expression
1 epfrc.1 . . 3
21frc 4541 . 2
3 dfin5 3321 . . . . 5
4 epel 4490 . . . . . . 7
54a1i 11 . . . . . 6
65rabbiia 2939 . . . . 5
73, 6eqtr4i 2459 . . . 4
87eqeq1i 2443 . . 3
98rexbii 2723 . 2
102, 9sylibr 204 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   w3a 936   wceq 1652   wcel 1725   wne 2599  wrex 2699  crab 2702  cvv 2949   cin 3312   wss 3313  c0 3621   class class class wbr 4205   cep 4485   wfr 4531 This theorem is referenced by:  wefrc  4569  onfr  4613  epfrs  7660 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 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-br 4206  df-opab 4260  df-eprel 4487  df-fr 4534
 Copyright terms: Public domain W3C validator