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Theorem epfrc 4502
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 4482 . 2  |-  ( (  _E  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  { y  e.  B  |  y  _E  x }  =  (/) )
3 dfin5 3264 . . . . 5  |-  ( B  i^i  x )  =  { y  e.  B  |  y  e.  x }
4 epel 4431 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
54a1i 11 . . . . . 6  |-  ( y  e.  B  ->  (
y  _E  x  <->  y  e.  x ) )
65rabbiia 2882 . . . . 5  |-  { y  e.  B  |  y  _E  x }  =  { y  e.  B  |  y  e.  x }
73, 6eqtr4i 2403 . . . 4  |-  ( B  i^i  x )  =  { y  e.  B  |  y  _E  x }
87eqeq1i 2387 . . 3  |-  ( ( B  i^i  x )  =  (/)  <->  { y  e.  B  |  y  _E  x }  =  (/) )
98rexbii 2667 . 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 1649    e. wcel 1717    =/= wne 2543   E.wrex 2643   {crab 2646   _Vcvv 2892    i^i cin 3255    C_ wss 3256   (/)c0 3564   class class class wbr 4146    _E cep 4426    Fr wfr 4472
This theorem is referenced by:  wefrc  4510  onfr  4554  epfrs  7593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-eprel 4428  df-fr 4475
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