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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  |-  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 4541 . 2  |-  ( (  _E  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  { y  e.  B  |  y  _E  x }  =  (/) )
3 dfin5 3321 . . . . 5  |-  ( B  i^i  x )  =  { y  e.  B  |  y  e.  x }
4 epel 4490 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
54a1i 11 . . . . . 6  |-  ( y  e.  B  ->  (
y  _E  x  <->  y  e.  x ) )
65rabbiia 2939 . . . . 5  |-  { y  e.  B  |  y  _E  x }  =  { y  e.  B  |  y  e.  x }
73, 6eqtr4i 2459 . . . 4  |-  ( B  i^i  x )  =  { y  e.  B  |  y  _E  x }
87eqeq1i 2443 . . 3  |-  ( ( B  i^i  x )  =  (/)  <->  { y  e.  B  |  y  _E  x }  =  (/) )
98rexbii 2723 . 2  |-  ( E. x  e.  B  ( B  i^i  x )  =  (/)  <->  E. x  e.  B  { y  e.  B  |  y  _E  x }  =  (/) )
102, 9sylibr 204 1  |-  ( (  _E  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  ( B  i^i  x )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2699   {crab 2702   _Vcvv 2949    i^i cin 3312    C_ wss 3313   (/)c0 3621   class class class wbr 4205    _E cep 4485    Fr 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
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