MPE Home 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  |-  B  e. 
_V
Assertion
Ref Expression
epfrc  |-  ( (  _E  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  ( B  i^i  x )  =  (/) )
Distinct variable groups:    x, A    x, B

Proof of Theorem epfrc
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 epfrc.1 . . 3  |-  B  e. 
_V
21frc 4793 . 2  |-  ( (  _E  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  { y  e.  B  |  y  _E  x }  =  (/) )
3 dfin5 3443 . . . . 5  |-  ( B  i^i  x )  =  { y  e.  B  |  y  e.  x }
4 epel 4742 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
54a1i 11 . . . . . 6  |-  ( y  e.  B  ->  (
y  _E  x  <->  y  e.  x ) )
65rabbiia 3065 . . . . 5  |-  { y  e.  B  |  y  _E  x }  =  { y  e.  B  |  y  e.  x }
73, 6eqtr4i 2486 . . . 4  |-  ( B  i^i  x )  =  { y  e.  B  |  y  _E  x }
87eqeq1i 2461 . . 3  |-  ( ( B  i^i  x )  =  (/)  <->  { y  e.  B  |  y  _E  x }  =  (/) )
98rexbii 2858 . 2  |-  ( E. x  e.  B  ( B  i^i  x )  =  (/)  <->  E. x  e.  B  { y  e.  B  |  y  _E  x }  =  (/) )
102, 9sylibr 212 1  |-  ( (  _E  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  ( B  i^i  x )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   E.wrex 2799   {crab 2802   _Vcvv 3076    i^i cin 3434    C_ wss 3435   (/)c0 3744   class class class wbr 4399    _E cep 4737    Fr 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