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Theorem suppr 7997
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 1005 . 2  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  R  Or  A )
2 ifcl 3953 . . 3  |-  ( ( B  e.  A  /\  C  e.  A )  ->  if ( C R B ,  B ,  C )  e.  A
)
323adant1 1023 . 2  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  if ( C R B ,  B ,  C )  e.  A
)
4 ifpr 4048 . . 3  |-  ( ( B  e.  A  /\  C  e.  A )  ->  if ( C R B ,  B ,  C )  e.  { B ,  C }
)
543adant1 1023 . 2  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  if ( C R B ,  B ,  C )  e.  { B ,  C }
)
6 breq1 4426 . . . . . 6  |-  ( B  =  if ( C R B ,  B ,  C )  ->  ( B R B  <->  if ( C R B ,  B ,  C ) R B ) )
76notbid 295 . . . . 5  |-  ( B  =  if ( C R B ,  B ,  C )  ->  ( -.  B R B  <->  -.  if ( C R B ,  B ,  C ) R B ) )
8 breq1 4426 . . . . . 6  |-  ( C  =  if ( C R B ,  B ,  C )  ->  ( C R B  <->  if ( C R B ,  B ,  C ) R B ) )
98notbid 295 . . . . 5  |-  ( C  =  if ( C R B ,  B ,  C )  ->  ( -.  C R B  <->  -.  if ( C R B ,  B ,  C ) R B ) )
10 sonr 4795 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
11103adant3 1025 . . . . . 6  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  B R B )
1211adantr 466 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  C R B )  ->  -.  B R B )
13 simpr 462 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  -.  C R B )  ->  -.  C R B )
147, 9, 12, 13ifbothda 3946 . . . 4  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  if ( C R B ,  B ,  C ) R B )
15 breq1 4426 . . . . . 6  |-  ( B  =  if ( C R B ,  B ,  C )  ->  ( B R C  <->  if ( C R B ,  B ,  C ) R C ) )
1615notbid 295 . . . . 5  |-  ( B  =  if ( C R B ,  B ,  C )  ->  ( -.  B R C  <->  -.  if ( C R B ,  B ,  C ) R C ) )
17 breq1 4426 . . . . . 6  |-  ( C  =  if ( C R B ,  B ,  C )  ->  ( C R C  <->  if ( C R B ,  B ,  C ) R C ) )
1817notbid 295 . . . . 5  |-  ( C  =  if ( C R B ,  B ,  C )  ->  ( -.  C R C  <->  -.  if ( C R B ,  B ,  C ) R C ) )
19 so2nr 4798 . . . . . . . . 9  |-  ( ( R  Or  A  /\  ( C  e.  A  /\  B  e.  A
) )  ->  -.  ( C R B  /\  B R C ) )
20193impb 1201 . . . . . . . 8  |-  ( ( R  Or  A  /\  C  e.  A  /\  B  e.  A )  ->  -.  ( C R B  /\  B R C ) )
21203com23 1211 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  ( C R B  /\  B R C ) )
22 imnan 423 . . . . . . 7  |-  ( ( C R B  ->  -.  B R C )  <->  -.  ( C R B  /\  B R C ) )
2321, 22sylibr 215 . . . . . 6  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  ( C R B  ->  -.  B R C ) )
2423imp 430 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  C R B )  ->  -.  B R C )
25 sonr 4795 . . . . . . 7  |-  ( ( R  Or  A  /\  C  e.  A )  ->  -.  C R C )
26253adant2 1024 . . . . . 6  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  C R C )
2726adantr 466 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  -.  C R B )  ->  -.  C R C )
2816, 18, 24, 27ifbothda 3946 . . . 4  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  if ( C R B ,  B ,  C ) R C )
29 breq2 4427 . . . . . . 7  |-  ( y  =  B  ->  ( if ( C R B ,  B ,  C
) R y  <->  if ( C R B ,  B ,  C ) R B ) )
3029notbid 295 . . . . . 6  |-  ( y  =  B  ->  ( -.  if ( C R B ,  B ,  C ) R y  <->  -.  if ( C R B ,  B ,  C ) R B ) )
31 breq2 4427 . . . . . . 7  |-  ( y  =  C  ->  ( if ( C R B ,  B ,  C
) R y  <->  if ( C R B ,  B ,  C ) R C ) )
3231notbid 295 . . . . . 6  |-  ( y  =  C  ->  ( -.  if ( C R B ,  B ,  C ) R y  <->  -.  if ( C R B ,  B ,  C ) R C ) )
3330, 32ralprg 4049 . . . . 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 1023 . . . 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 930 . . 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 2791 . 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 7991 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2771   ifcif 3911   {cpr 4000   class class class wbr 4423    Or wor 4773   supcsup 7964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-po 4774  df-so 4775  df-iota 5565  df-riota 6268  df-sup 7966
This theorem is referenced by:  supsn  7998  tmsxpsval2  21553  esumsnf  28894  sge0sn  38130
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