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Theorem suppr 7932
Description: The supremum of a pair. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
suppr  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  sup ( { B ,  C } ,  A ,  R )  =  if ( C R B ,  B ,  C
) )

Proof of Theorem suppr
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simp1 997 . 2  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  R  Or  A )
2 ifcl 3968 . . 3  |-  ( ( B  e.  A  /\  C  e.  A )  ->  if ( C R B ,  B ,  C )  e.  A
)
323adant1 1015 . 2  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  if ( C R B ,  B ,  C )  e.  A
)
4 ifpr 4062 . . 3  |-  ( ( B  e.  A  /\  C  e.  A )  ->  if ( C R B ,  B ,  C )  e.  { B ,  C }
)
543adant1 1015 . 2  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  if ( C R B ,  B ,  C )  e.  { B ,  C }
)
6 breq1 4440 . . . . . 6  |-  ( B  =  if ( C R B ,  B ,  C )  ->  ( B R B  <->  if ( C R B ,  B ,  C ) R B ) )
76notbid 294 . . . . 5  |-  ( B  =  if ( C R B ,  B ,  C )  ->  ( -.  B R B  <->  -.  if ( C R B ,  B ,  C ) R B ) )
8 breq1 4440 . . . . . 6  |-  ( C  =  if ( C R B ,  B ,  C )  ->  ( C R B  <->  if ( C R B ,  B ,  C ) R B ) )
98notbid 294 . . . . 5  |-  ( C  =  if ( C R B ,  B ,  C )  ->  ( -.  C R B  <->  -.  if ( C R B ,  B ,  C ) R B ) )
10 sonr 4811 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
11103adant3 1017 . . . . . 6  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  B R B )
1211adantr 465 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  C R B )  ->  -.  B R B )
13 simpr 461 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  -.  C R B )  ->  -.  C R B )
147, 9, 12, 13ifbothda 3961 . . . 4  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  if ( C R B ,  B ,  C ) R B )
15 breq1 4440 . . . . . 6  |-  ( B  =  if ( C R B ,  B ,  C )  ->  ( B R C  <->  if ( C R B ,  B ,  C ) R C ) )
1615notbid 294 . . . . 5  |-  ( B  =  if ( C R B ,  B ,  C )  ->  ( -.  B R C  <->  -.  if ( C R B ,  B ,  C ) R C ) )
17 breq1 4440 . . . . . 6  |-  ( C  =  if ( C R B ,  B ,  C )  ->  ( C R C  <->  if ( C R B ,  B ,  C ) R C ) )
1817notbid 294 . . . . 5  |-  ( C  =  if ( C R B ,  B ,  C )  ->  ( -.  C R C  <->  -.  if ( C R B ,  B ,  C ) R C ) )
19 so2nr 4814 . . . . . . . . 9  |-  ( ( R  Or  A  /\  ( C  e.  A  /\  B  e.  A
) )  ->  -.  ( C R B  /\  B R C ) )
20193impb 1193 . . . . . . . 8  |-  ( ( R  Or  A  /\  C  e.  A  /\  B  e.  A )  ->  -.  ( C R B  /\  B R C ) )
21203com23 1203 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  ( C R B  /\  B R C ) )
22 imnan 422 . . . . . . 7  |-  ( ( C R B  ->  -.  B R C )  <->  -.  ( C R B  /\  B R C ) )
2321, 22sylibr 212 . . . . . 6  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  ( C R B  ->  -.  B R C ) )
2423imp 429 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  C R B )  ->  -.  B R C )
25 sonr 4811 . . . . . . 7  |-  ( ( R  Or  A  /\  C  e.  A )  ->  -.  C R C )
26253adant2 1016 . . . . . 6  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  C R C )
2726adantr 465 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  -.  C R B )  ->  -.  C R C )
2816, 18, 24, 27ifbothda 3961 . . . 4  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  if ( C R B ,  B ,  C ) R C )
29 breq2 4441 . . . . . . 7  |-  ( y  =  B  ->  ( if ( C R B ,  B ,  C
) R y  <->  if ( C R B ,  B ,  C ) R B ) )
3029notbid 294 . . . . . 6  |-  ( y  =  B  ->  ( -.  if ( C R B ,  B ,  C ) R y  <->  -.  if ( C R B ,  B ,  C ) R B ) )
31 breq2 4441 . . . . . . 7  |-  ( y  =  C  ->  ( if ( C R B ,  B ,  C
) R y  <->  if ( C R B ,  B ,  C ) R C ) )
3231notbid 294 . . . . . 6  |-  ( y  =  C  ->  ( -.  if ( C R B ,  B ,  C ) R y  <->  -.  if ( C R B ,  B ,  C ) R C ) )
3330, 32ralprg 4063 . . . . 5  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( A. y  e. 
{ B ,  C }  -.  if ( C R B ,  B ,  C ) R y  <-> 
( -.  if ( C R B ,  B ,  C ) R B  /\  -.  if ( C R B ,  B ,  C ) R C ) ) )
34333adant1 1015 . . . 4  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  ( A. y  e. 
{ B ,  C }  -.  if ( C R B ,  B ,  C ) R y  <-> 
( -.  if ( C R B ,  B ,  C ) R B  /\  -.  if ( C R B ,  B ,  C ) R C ) ) )
3514, 28, 34mpbir2and 922 . . 3  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  A. y  e.  { B ,  C }  -.  if ( C R B ,  B ,  C ) R y )
3635r19.21bi 2812 . 2  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  y  e.  { B ,  C }
)  ->  -.  if ( C R B ,  B ,  C ) R y )
371, 3, 5, 36supmax 7926 1  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  sup ( { B ,  C } ,  A ,  R )  =  if ( C R B ,  B ,  C
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   ifcif 3926   {cpr 4016   class class class wbr 4437    Or wor 4789   supcsup 7902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-po 4790  df-so 4791  df-iota 5541  df-riota 6242  df-sup 7903
This theorem is referenced by:  supsn  7933  tmsxpsval2  21020  esumsn  28050
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