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Theorem epfrc 4395
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 4375 . 2  |-  ( (  _E  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  { y  e.  B  |  y  _E  x }  =  (/) )
3 dfin5 3173 . . . . 5  |-  ( B  i^i  x )  =  { y  e.  B  |  y  e.  x }
4 epel 4324 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
54a1i 10 . . . . . 6  |-  ( y  e.  B  ->  (
y  _E  x  <->  y  e.  x ) )
65rabbiia 2791 . . . . 5  |-  { y  e.  B  |  y  _E  x }  =  { y  e.  B  |  y  e.  x }
73, 6eqtr4i 2319 . . . 4  |-  ( B  i^i  x )  =  { y  e.  B  |  y  _E  x }
87eqeq1i 2303 . . 3  |-  ( ( B  i^i  x )  =  (/)  <->  { y  e.  B  |  y  _E  x }  =  (/) )
98rexbii 2581 . 2  |-  ( E. x  e.  B  ( B  i^i  x )  =  (/)  <->  E. x  e.  B  { y  e.  B  |  y  _E  x }  =  (/) )
102, 9sylibr 203 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 176    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468   class class class wbr 4039    _E cep 4319    Fr wfr 4365
This theorem is referenced by:  wefrc  4403  onfr  4447  epfrs  7429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-eprel 4321  df-fr 4368
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