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

Theorem epfrc 4813
 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 4793 . 2
3 dfin5 3443 . . . . 5
4 epel 4742 . . . . . . 7
54a1i 11 . . . . . 6
65rabbiia 3065 . . . . 5
73, 6eqtr4i 2486 . . . 4
87eqeq1i 2461 . . 3
98rexbii 2858 . 2
102, 9sylibr 212 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   w3a 965   wceq 1370   wcel 1758   wne 2647  wrex 2799  crab 2802  cvv 3076   cin 3434   wss 3435  c0 3744   class class class wbr 4399   cep 4737   wfr 4783 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-br 4400  df-opab 4458  df-eprel 4739  df-fr 4786 This theorem is referenced by:  wefrc  4821  onfr  4865  epfrs  8061
 Copyright terms: Public domain W3C validator