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Theorem somin2 5212
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 5211 . 2  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  if ( A R B ,  A ,  B )  =  if ( B R A ,  B ,  A ) )
2 somin1 5210 . . 3  |-  ( ( R  Or  X  /\  ( B  e.  X  /\  A  e.  X
) )  ->  if ( B R A ,  B ,  A )
( R  u.  _I  ) B )
32ancom2s 778 . 2  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  if ( B R A ,  B ,  A )
( R  u.  _I  ) B )
41, 3eqbrtrd 4173 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 359    e. wcel 1717    u. cun 3261   ifcif 3682   class class class wbr 4153    _I cid 4434    Or wor 4443
This theorem is referenced by:  soltmin  5213
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 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825
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