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Theorem soltmin 5232
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 4480 . . . . . 6  |-  ( R  Or  X  ->  R  Po  X )
21ad2antrr 707 . . . . 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 999 . . . . . 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 1000 . . . . . . 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 1001 . . . . . . 7  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  C  e.  X )
6 ifcl 3735 . . . . . . 7  |-  ( ( B  e.  X  /\  C  e.  X )  ->  if ( B R C ,  B ,  C )  e.  X
)
74, 5, 6syl2anc 643 . . . . . 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 )
83, 7, 43jca 1134 . . . . 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
) )
9 simpr 448 . . . . 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 ) )
10 simpll 731 . . . . . 6  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  R  Or  X )
11 somin1 5229 . . . . . 6  |-  ( ( R  Or  X  /\  ( B  e.  X  /\  C  e.  X
) )  ->  if ( B R C ,  B ,  C )
( R  u.  _I  ) B )
1210, 4, 5, 11syl12anc 1182 . . . . 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 )
13 poltletr 5228 . . . . . 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 ) )
1413imp 419 . . . . 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 )
152, 8, 9, 12, 14syl22anc 1185 . . . 4  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  A R B )
163, 7, 53jca 1134 . . . . 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
) )
17 somin2 5231 . . . . . 6  |-  ( ( R  Or  X  /\  ( B  e.  X  /\  C  e.  X
) )  ->  if ( B R C ,  B ,  C )
( R  u.  _I  ) C )
1810, 4, 5, 17syl12anc 1182 . . . . 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 )
19 poltletr 5228 . . . . . 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 ) )
2019imp 419 . . . . 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 )
212, 16, 9, 18, 20syl22anc 1185 . . . 4  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  A R C )
2215, 21jca 519 . . 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 ) )
2322ex 424 . 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 ) ) )
24 breq2 4176 . . 3  |-  ( B  =  if ( B R C ,  B ,  C )  ->  ( A R B  <->  A R if ( B R C ,  B ,  C
) ) )
25 breq2 4176 . . 3  |-  ( C  =  if ( B R C ,  B ,  C )  ->  ( A R C  <->  A R if ( B R C ,  B ,  C
) ) )
2624, 25ifboth 3730 . 2  |-  ( ( A R B  /\  A R C )  ->  A R if ( B R C ,  B ,  C ) )
2723, 26impbid1 195 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1721    u. cun 3278   ifcif 3699   class class class wbr 4172    _I cid 4453    Po wpo 4461    Or wor 4462
This theorem is referenced by:  wemaplem2  7472
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844
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