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Theorem suppr 7235
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 variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 955 . 2  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  R  Or  A )
2 ifcl 3614 . . 3  |-  ( ( B  e.  A  /\  C  e.  A )  ->  if ( C R B ,  B ,  C )  e.  A
)
323adant1 973 . 2  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  if ( C R B ,  B ,  C )  e.  A
)
4 breq1 4042 . . . . . 6  |-  ( B  =  if ( C R B ,  B ,  C )  ->  ( B R B  <->  if ( C R B ,  B ,  C ) R B ) )
54notbid 285 . . . . 5  |-  ( B  =  if ( C R B ,  B ,  C )  ->  ( -.  B R B  <->  -.  if ( C R B ,  B ,  C ) R B ) )
6 breq1 4042 . . . . . 6  |-  ( C  =  if ( C R B ,  B ,  C )  ->  ( C R B  <->  if ( C R B ,  B ,  C ) R B ) )
76notbid 285 . . . . 5  |-  ( C  =  if ( C R B ,  B ,  C )  ->  ( -.  C R B  <->  -.  if ( C R B ,  B ,  C ) R B ) )
8 sonr 4351 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
983adant3 975 . . . . . 6  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  B R B )
109adantr 451 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  C R B )  ->  -.  B R B )
11 simpr 447 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  -.  C R B )  ->  -.  C R B )
125, 7, 10, 11ifbothda 3608 . . . 4  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  if ( C R B ,  B ,  C ) R B )
13 breq1 4042 . . . . . 6  |-  ( B  =  if ( C R B ,  B ,  C )  ->  ( B R C  <->  if ( C R B ,  B ,  C ) R C ) )
1413notbid 285 . . . . 5  |-  ( B  =  if ( C R B ,  B ,  C )  ->  ( -.  B R C  <->  -.  if ( C R B ,  B ,  C ) R C ) )
15 breq1 4042 . . . . . 6  |-  ( C  =  if ( C R B ,  B ,  C )  ->  ( C R C  <->  if ( C R B ,  B ,  C ) R C ) )
1615notbid 285 . . . . 5  |-  ( C  =  if ( C R B ,  B ,  C )  ->  ( -.  C R C  <->  -.  if ( C R B ,  B ,  C ) R C ) )
17 so2nr 4354 . . . . . . . . 9  |-  ( ( R  Or  A  /\  ( C  e.  A  /\  B  e.  A
) )  ->  -.  ( C R B  /\  B R C ) )
18173impb 1147 . . . . . . . 8  |-  ( ( R  Or  A  /\  C  e.  A  /\  B  e.  A )  ->  -.  ( C R B  /\  B R C ) )
19183com23 1157 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  ( C R B  /\  B R C ) )
20 imnan 411 . . . . . . 7  |-  ( ( C R B  ->  -.  B R C )  <->  -.  ( C R B  /\  B R C ) )
2119, 20sylibr 203 . . . . . 6  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  ( C R B  ->  -.  B R C ) )
2221imp 418 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  C R B )  ->  -.  B R C )
23 sonr 4351 . . . . . . 7  |-  ( ( R  Or  A  /\  C  e.  A )  ->  -.  C R C )
24233adant2 974 . . . . . 6  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  C R C )
2524adantr 451 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  -.  C R B )  ->  -.  C R C )
2614, 16, 22, 25ifbothda 3608 . . . 4  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  if ( C R B ,  B ,  C ) R C )
27 breq2 4043 . . . . . . 7  |-  ( y  =  B  ->  ( if ( C R B ,  B ,  C
) R y  <->  if ( C R B ,  B ,  C ) R B ) )
2827notbid 285 . . . . . 6  |-  ( y  =  B  ->  ( -.  if ( C R B ,  B ,  C ) R y  <->  -.  if ( C R B ,  B ,  C ) R B ) )
29 breq2 4043 . . . . . . 7  |-  ( y  =  C  ->  ( if ( C R B ,  B ,  C
) R y  <->  if ( C R B ,  B ,  C ) R C ) )
3029notbid 285 . . . . . 6  |-  ( y  =  C  ->  ( -.  if ( C R B ,  B ,  C ) R y  <->  -.  if ( C R B ,  B ,  C ) R C ) )
3128, 30ralprg 3695 . . . . 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 ) ) )
32313adant1 973 . . . 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 ) ) )
3312, 26, 32mpbir2and 888 . . 3  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  A. y  e.  { B ,  C }  -.  if ( C R B ,  B ,  C ) R y )
3433r19.21bi 2654 . 2  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  y  e.  { B ,  C }
)  ->  -.  if ( C R B ,  B ,  C ) R y )
35 ifpr 3694 . . . . . 6  |-  ( ( B  e.  A  /\  C  e.  A )  ->  if ( C R B ,  B ,  C )  e.  { B ,  C }
)
36353adant1 973 . . . . 5  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  if ( C R B ,  B ,  C )  e.  { B ,  C }
)
3736adantr 451 . . . 4  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  y  e.  A )  ->  if ( C R B ,  B ,  C )  e.  { B ,  C } )
38 breq2 4043 . . . . 5  |-  ( z  =  if ( C R B ,  B ,  C )  ->  (
y R z  <->  y R if ( C R B ,  B ,  C
) ) )
3938rspcev 2897 . . . 4  |-  ( ( if ( C R B ,  B ,  C )  e.  { B ,  C }  /\  y R if ( C R B ,  B ,  C )
)  ->  E. z  e.  { B ,  C } y R z )
4037, 39sylan 457 . . 3  |-  ( ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  y  e.  A )  /\  y R if ( C R B ,  B ,  C ) )  ->  E. z  e.  { B ,  C } y R z )
4140anasss 628 . 2  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  ( y  e.  A  /\  y R if ( C R B ,  B ,  C ) ) )  ->  E. z  e.  { B ,  C }
y R z )
421, 3, 34, 41eqsupd 7224 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   ifcif 3578   {cpr 3654   class class class wbr 4039    Or wor 4329   supcsup 7209
This theorem is referenced by:  supsn  7236  tmsxpsval2  18101  esumsn  23452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-po 4330  df-so 4331  df-iota 5235  df-riota 6320  df-sup 7210
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