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Theorem epfrc 4779
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 4759 . 2  |-  ( (  _E  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  { y  e.  B  |  y  _E  x }  =  (/) )
3 dfin5 3397 . . . . 5  |-  ( B  i^i  x )  =  { y  e.  B  |  y  e.  x }
4 epel 4708 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
54a1i 11 . . . . . 6  |-  ( y  e.  B  ->  (
y  _E  x  <->  y  e.  x ) )
65rabbiia 3023 . . . . 5  |-  { y  e.  B  |  y  _E  x }  =  { y  e.  B  |  y  e.  x }
73, 6eqtr4i 2414 . . . 4  |-  ( B  i^i  x )  =  { y  e.  B  |  y  _E  x }
87eqeq1i 2389 . . 3  |-  ( ( B  i^i  x )  =  (/)  <->  { y  e.  B  |  y  _E  x }  =  (/) )
98rexbii 2884 . 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 971    = wceq 1399    e. wcel 1826    =/= wne 2577   E.wrex 2733   {crab 2736   _Vcvv 3034    i^i cin 3388    C_ wss 3389   (/)c0 3711   class class class wbr 4367    _E cep 4703    Fr wfr 4749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-eprel 4705  df-fr 4752
This theorem is referenced by:  wefrc  4787  onfr  4831  epfrs  8075
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