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Theorem soltmin 5391
Description: Being less than a minimum, for a general total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
soltmin  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A R if ( B R C ,  B ,  C )  <->  ( A R B  /\  A R C ) ) )

Proof of Theorem soltmin
StepHypRef Expression
1 sopo 4806 . . . . . 6  |-  ( R  Or  X  ->  R  Po  X )
21ad2antrr 723 . . . . 5  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  R  Po  X )
3 simplr1 1036 . . . . . 6  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  A  e.  X )
4 simplr2 1037 . . . . . . 7  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  B  e.  X )
5 simplr3 1038 . . . . . . 7  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  C  e.  X )
64, 5ifcld 3972 . . . . . 6  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  if ( B R C ,  B ,  C
)  e.  X )
73, 6, 43jca 1174 . . . . 5  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  -> 
( A  e.  X  /\  if ( B R C ,  B ,  C )  e.  X  /\  B  e.  X
) )
8 simpr 459 . . . . 5  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  A R if ( B R C ,  B ,  C ) )
9 simpll 751 . . . . . 6  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  R  Or  X )
10 somin1 5388 . . . . . 6  |-  ( ( R  Or  X  /\  ( B  e.  X  /\  C  e.  X
) )  ->  if ( B R C ,  B ,  C )
( R  u.  _I  ) B )
119, 4, 5, 10syl12anc 1224 . . . . 5  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  if ( B R C ,  B ,  C
) ( R  u.  _I  ) B )
12 poltletr 5387 . . . . . 6  |-  ( ( R  Po  X  /\  ( A  e.  X  /\  if ( B R C ,  B ,  C )  e.  X  /\  B  e.  X
) )  ->  (
( A R if ( B R C ,  B ,  C
)  /\  if ( B R C ,  B ,  C ) ( R  u.  _I  ) B )  ->  A R B ) )
1312imp 427 . . . . 5  |-  ( ( ( R  Po  X  /\  ( A  e.  X  /\  if ( B R C ,  B ,  C )  e.  X  /\  B  e.  X
) )  /\  ( A R if ( B R C ,  B ,  C )  /\  if ( B R C ,  B ,  C )
( R  u.  _I  ) B ) )  ->  A R B )
142, 7, 8, 11, 13syl22anc 1227 . . . 4  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  A R B )
153, 6, 53jca 1174 . . . . 5  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  -> 
( A  e.  X  /\  if ( B R C ,  B ,  C )  e.  X  /\  C  e.  X
) )
16 somin2 5390 . . . . . 6  |-  ( ( R  Or  X  /\  ( B  e.  X  /\  C  e.  X
) )  ->  if ( B R C ,  B ,  C )
( R  u.  _I  ) C )
179, 4, 5, 16syl12anc 1224 . . . . 5  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  if ( B R C ,  B ,  C
) ( R  u.  _I  ) C )
18 poltletr 5387 . . . . . 6  |-  ( ( R  Po  X  /\  ( A  e.  X  /\  if ( B R C ,  B ,  C )  e.  X  /\  C  e.  X
) )  ->  (
( A R if ( B R C ,  B ,  C
)  /\  if ( B R C ,  B ,  C ) ( R  u.  _I  ) C )  ->  A R C ) )
1918imp 427 . . . . 5  |-  ( ( ( R  Po  X  /\  ( A  e.  X  /\  if ( B R C ,  B ,  C )  e.  X  /\  C  e.  X
) )  /\  ( A R if ( B R C ,  B ,  C )  /\  if ( B R C ,  B ,  C )
( R  u.  _I  ) C ) )  ->  A R C )
202, 15, 8, 17, 19syl22anc 1227 . . . 4  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  A R C )
2114, 20jca 530 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  -> 
( A R B  /\  A R C ) )
2221ex 432 . 2  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A R if ( B R C ,  B ,  C )  ->  ( A R B  /\  A R C ) ) )
23 breq2 4443 . . 3  |-  ( B  =  if ( B R C ,  B ,  C )  ->  ( A R B  <->  A R if ( B R C ,  B ,  C
) ) )
24 breq2 4443 . . 3  |-  ( C  =  if ( B R C ,  B ,  C )  ->  ( A R C  <->  A R if ( B R C ,  B ,  C
) ) )
2523, 24ifboth 3965 . 2  |-  ( ( A R B  /\  A R C )  ->  A R if ( B R C ,  B ,  C ) )
2622, 25impbid1 203 1  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A R if ( B R C ,  B ,  C )  <->  ( A R B  /\  A R C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    e. wcel 1823    u. cun 3459   ifcif 3929   class class class wbr 4439    _I cid 4779    Po wpo 4787    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:  wemaplem2  7964
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