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Theorem somin1 5079
Description: Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
somin1  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  if ( A R B ,  A ,  B )
( R  u.  _I  ) A )

Proof of Theorem somin1
StepHypRef Expression
1 iftrue 3571 . . . . 5  |-  ( A R B  ->  if ( A R B ,  A ,  B )  =  A )
21olcd 382 . . . 4  |-  ( A R B  ->  ( if ( A R B ,  A ,  B
) R A  \/  if ( A R B ,  A ,  B
)  =  A ) )
32adantl 452 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  ( if ( A R B ,  A ,  B
) R A  \/  if ( A R B ,  A ,  B
)  =  A ) )
4 sotric 4340 . . . . . . 7  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( A R B  <->  -.  ( A  =  B  \/  B R A ) ) )
5 orcom 376 . . . . . . . . 9  |-  ( ( A  =  B  \/  B R A )  <->  ( B R A  \/  A  =  B ) )
6 eqcom 2285 . . . . . . . . . 10  |-  ( A  =  B  <->  B  =  A )
76orbi2i 505 . . . . . . . . 9  |-  ( ( B R A  \/  A  =  B )  <->  ( B R A  \/  B  =  A )
)
85, 7bitri 240 . . . . . . . 8  |-  ( ( A  =  B  \/  B R A )  <->  ( B R A  \/  B  =  A ) )
98notbii 287 . . . . . . 7  |-  ( -.  ( A  =  B  \/  B R A )  <->  -.  ( B R A  \/  B  =  A ) )
104, 9syl6bb 252 . . . . . 6  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( A R B  <->  -.  ( B R A  \/  B  =  A ) ) )
1110con2bid 319 . . . . 5  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( B R A  \/  B  =  A )  <->  -.  A R B ) )
1211biimpar 471 . . . 4  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  -.  A R B )  -> 
( B R A  \/  B  =  A ) )
13 iffalse 3572 . . . . . 6  |-  ( -.  A R B  ->  if ( A R B ,  A ,  B
)  =  B )
14 breq1 4026 . . . . . . 7  |-  ( if ( A R B ,  A ,  B
)  =  B  -> 
( if ( A R B ,  A ,  B ) R A  <-> 
B R A ) )
15 eqeq1 2289 . . . . . . 7  |-  ( if ( A R B ,  A ,  B
)  =  B  -> 
( if ( A R B ,  A ,  B )  =  A  <-> 
B  =  A ) )
1614, 15orbi12d 690 . . . . . 6  |-  ( if ( A R B ,  A ,  B
)  =  B  -> 
( ( if ( A R B ,  A ,  B ) R A  \/  if ( A R B ,  A ,  B )  =  A )  <->  ( B R A  \/  B  =  A ) ) )
1713, 16syl 15 . . . . 5  |-  ( -.  A R B  -> 
( ( if ( A R B ,  A ,  B ) R A  \/  if ( A R B ,  A ,  B )  =  A )  <->  ( B R A  \/  B  =  A ) ) )
1817adantl 452 . . . 4  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  -.  A R B )  -> 
( ( if ( A R B ,  A ,  B ) R A  \/  if ( A R B ,  A ,  B )  =  A )  <->  ( B R A  \/  B  =  A ) ) )
1912, 18mpbird 223 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  -.  A R B )  -> 
( if ( A R B ,  A ,  B ) R A  \/  if ( A R B ,  A ,  B )  =  A ) )
203, 19pm2.61dan 766 . 2  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( if ( A R B ,  A ,  B
) R A  \/  if ( A R B ,  A ,  B
)  =  A ) )
21 poleloe 5077 . . 3  |-  ( A  e.  X  ->  ( if ( A R B ,  A ,  B
) ( R  u.  _I  ) A  <->  ( if ( A R B ,  A ,  B ) R A  \/  if ( A R B ,  A ,  B )  =  A ) ) )
2221ad2antrl 708 . 2  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( if ( A R B ,  A ,  B
) ( R  u.  _I  ) A  <->  ( if ( A R B ,  A ,  B ) R A  \/  if ( A R B ,  A ,  B )  =  A ) ) )
2320, 22mpbird 223 1  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  if ( A R B ,  A ,  B )
( R  u.  _I  ) A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    u. cun 3150   ifcif 3565   class class class wbr 4023    _I cid 4304    Or wor 4313
This theorem is referenced by:  somin2  5081  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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