MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  somin1 Structured version   Unicode version

Theorem somin1 5337
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 3900 . . . . 5  |-  ( A R B  ->  if ( A R B ,  A ,  B )  =  A )
21olcd 393 . . . 4  |-  ( A R B  ->  ( if ( A R B ,  A ,  B
) R A  \/  if ( A R B ,  A ,  B
)  =  A ) )
32adantl 466 . . 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 4770 . . . . . . 7  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( A R B  <->  -.  ( A  =  B  \/  B R A ) ) )
5 orcom 387 . . . . . . . . 9  |-  ( ( A  =  B  \/  B R A )  <->  ( B R A  \/  A  =  B ) )
6 eqcom 2461 . . . . . . . . . 10  |-  ( A  =  B  <->  B  =  A )
76orbi2i 519 . . . . . . . . 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 296 . . . . . . 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 329 . . . . 5  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( B R A  \/  B  =  A )  <->  -.  A R B ) )
1211biimpar 485 . . . 4  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  -.  A R B )  -> 
( B R A  \/  B  =  A ) )
13 iffalse 3902 . . . . . 6  |-  ( -.  A R B  ->  if ( A R B ,  A ,  B
)  =  B )
14 breq1 4398 . . . . . . 7  |-  ( if ( A R B ,  A ,  B
)  =  B  -> 
( if ( A R B ,  A ,  B ) R A  <-> 
B R A ) )
15 eqeq1 2456 . . . . . . 7  |-  ( if ( A R B ,  A ,  B
)  =  B  -> 
( if ( A R B ,  A ,  B )  =  A  <-> 
B  =  A ) )
1614, 15orbi12d 709 . . . . . 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 466 . . . 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 5335 . . 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 727 . 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 368    /\ wa 369    = wceq 1370    e. wcel 1758    u. cun 3429   ifcif 3894   class class class wbr 4395    _I cid 4734    Or wor 4743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-opab 4454  df-id 4739  df-po 4744  df-so 4745  df-xp 4949  df-rel 4950
This theorem is referenced by:  somin2  5339  soltmin  5340
  Copyright terms: Public domain W3C validator