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Theorem frminexOLD 26532
Description: If an element of a founded set satisfies a property  ph, then there is a minimal element that satisfies  ph. (Moved to frminex 4389 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Mar-2013.) (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
frminexOLD.1  |-  A  e. 
_V
frminexOLD.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
frminexOLD  |-  ( R  Fr  A  ->  ( E. x  e.  A  ph 
->  E. x  e.  A  ( ph  /\  A. y  e.  A  ( ps  ->  -.  y R x ) ) ) )
Distinct variable groups:    x, A, y    x, R, y    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem frminexOLD
StepHypRef Expression
1 frminexOLD.1 . 2  |-  A  e. 
_V
2 frminexOLD.2 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
31, 2frminex 4389 1  |-  ( R  Fr  A  ->  ( E. x  e.  A  ph 
->  E. x  e.  A  ( ph  /\  A. y  e.  A  ( ps  ->  -.  y R x ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801   class class class wbr 4039    Fr wfr 4365
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
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-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-fr 4368
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