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Theorem somin2 5081
Description: Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
somin2  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  if ( A R B ,  A ,  B )
( R  u.  _I  ) B )

Proof of Theorem somin2
StepHypRef Expression
1 somincom 5080 . 2  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  if ( A R B ,  A ,  B )  =  if ( B R A ,  B ,  A ) )
2 somin1 5079 . . 3  |-  ( ( R  Or  X  /\  ( B  e.  X  /\  A  e.  X
) )  ->  if ( B R A ,  B ,  A )
( R  u.  _I  ) B )
32ancom2s 777 . 2  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  if ( B R A ,  B ,  A )
( R  u.  _I  ) B )
41, 3eqbrtrd 4043 1  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  if ( A R B ,  A ,  B )
( R  u.  _I  ) B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684    u. cun 3150   ifcif 3565   class class class wbr 4023    _I cid 4304    Or wor 4313
This theorem is referenced by:  soltmin  5082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696
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