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Theorem somin1 5388
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 3935 . . . . 5  |-  ( A R B  ->  if ( A R B ,  A ,  B )  =  A )
21olcd 391 . . . 4  |-  ( A R B  ->  ( if ( A R B ,  A ,  B
) R A  \/  if ( A R B ,  A ,  B
)  =  A ) )
32adantl 464 . . 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 4815 . . . . . . 7  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( A R B  <->  -.  ( A  =  B  \/  B R A ) ) )
5 orcom 385 . . . . . . . . 9  |-  ( ( A  =  B  \/  B R A )  <->  ( B R A  \/  A  =  B ) )
6 eqcom 2463 . . . . . . . . . 10  |-  ( A  =  B  <->  B  =  A )
76orbi2i 517 . . . . . . . . 9  |-  ( ( B R A  \/  A  =  B )  <->  ( B R A  \/  B  =  A )
)
85, 7bitri 249 . . . . . . . 8  |-  ( ( A  =  B  \/  B R A )  <->  ( B R A  \/  B  =  A ) )
98notbii 294 . . . . . . 7  |-  ( -.  ( A  =  B  \/  B R A )  <->  -.  ( B R A  \/  B  =  A ) )
104, 9syl6bb 261 . . . . . 6  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( A R B  <->  -.  ( B R A  \/  B  =  A ) ) )
1110con2bid 327 . . . . 5  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( B R A  \/  B  =  A )  <->  -.  A R B ) )
1211biimpar 483 . . . 4  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  -.  A R B )  -> 
( B R A  \/  B  =  A ) )
13 iffalse 3938 . . . . . 6  |-  ( -.  A R B  ->  if ( A R B ,  A ,  B
)  =  B )
14 breq1 4442 . . . . . . 7  |-  ( if ( A R B ,  A ,  B
)  =  B  -> 
( if ( A R B ,  A ,  B ) R A  <-> 
B R A ) )
15 eqeq1 2458 . . . . . . 7  |-  ( if ( A R B ,  A ,  B
)  =  B  -> 
( if ( A R B ,  A ,  B )  =  A  <-> 
B  =  A ) )
1614, 15orbi12d 707 . . . . . 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 16 . . . . 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 464 . . . 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 232 . . 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 789 . 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 5386 . . 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 725 . 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 232 1  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  if ( A R B ,  A ,  B )
( R  u.  _I  ) A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    u. cun 3459   ifcif 3929   class class class wbr 4439    _I cid 4779    Or wor 4788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995
This theorem is referenced by:  somin2  5390  soltmin  5391
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