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Theorem supsnti 6807
Description: The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.)
Hypotheses
Ref Expression
supsnti.ti  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
supsnti.b  |-  ( ph  ->  B  e.  A )
Assertion
Ref Expression
supsnti  |-  ( ph  ->  sup ( { B } ,  A ,  R )  =  B )
Distinct variable groups:    u, A, v   
u, B, v    u, R, v    ph, u, v

Proof of Theorem supsnti
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 supsnti.ti . 2  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
2 supsnti.b . 2  |-  ( ph  ->  B  e.  A )
3 snidg 3501 . . 3  |-  ( B  e.  A  ->  B  e.  { B } )
42, 3syl 14 . 2  |-  ( ph  ->  B  e.  { B } )
5 eqid 2100 . . . . . 6  |-  B  =  B
61ralrimivva 2473 . . . . . . 7  |-  ( ph  ->  A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )
7 eqeq1 2106 . . . . . . . . . 10  |-  ( u  =  B  ->  (
u  =  v  <->  B  =  v ) )
8 breq1 3878 . . . . . . . . . . . 12  |-  ( u  =  B  ->  (
u R v  <->  B R
v ) )
98notbid 633 . . . . . . . . . . 11  |-  ( u  =  B  ->  ( -.  u R v  <->  -.  B R v ) )
10 breq2 3879 . . . . . . . . . . . 12  |-  ( u  =  B  ->  (
v R u  <->  v R B ) )
1110notbid 633 . . . . . . . . . . 11  |-  ( u  =  B  ->  ( -.  v R u  <->  -.  v R B ) )
129, 11anbi12d 460 . . . . . . . . . 10  |-  ( u  =  B  ->  (
( -.  u R v  /\  -.  v R u )  <->  ( -.  B R v  /\  -.  v R B ) ) )
137, 12bibi12d 234 . . . . . . . . 9  |-  ( u  =  B  ->  (
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) )  <-> 
( B  =  v  <-> 
( -.  B R v  /\  -.  v R B ) ) ) )
14 eqeq2 2109 . . . . . . . . . 10  |-  ( v  =  B  ->  ( B  =  v  <->  B  =  B ) )
15 breq2 3879 . . . . . . . . . . . 12  |-  ( v  =  B  ->  ( B R v  <->  B R B ) )
1615notbid 633 . . . . . . . . . . 11  |-  ( v  =  B  ->  ( -.  B R v  <->  -.  B R B ) )
17 breq1 3878 . . . . . . . . . . . 12  |-  ( v  =  B  ->  (
v R B  <->  B R B ) )
1817notbid 633 . . . . . . . . . . 11  |-  ( v  =  B  ->  ( -.  v R B  <->  -.  B R B ) )
1916, 18anbi12d 460 . . . . . . . . . 10  |-  ( v  =  B  ->  (
( -.  B R v  /\  -.  v R B )  <->  ( -.  B R B  /\  -.  B R B ) ) )
2014, 19bibi12d 234 . . . . . . . . 9  |-  ( v  =  B  ->  (
( B  =  v  <-> 
( -.  B R v  /\  -.  v R B ) )  <->  ( B  =  B  <->  ( -.  B R B  /\  -.  B R B ) ) ) )
2113, 20rspc2v 2756 . . . . . . . 8  |-  ( ( B  e.  A  /\  B  e.  A )  ->  ( A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) )  ->  ( B  =  B  <->  ( -.  B R B  /\  -.  B R B ) ) ) )
222, 2, 21syl2anc 406 . . . . . . 7  |-  ( ph  ->  ( A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) )  ->  ( B  =  B  <->  ( -.  B R B  /\  -.  B R B ) ) ) )
236, 22mpd 13 . . . . . 6  |-  ( ph  ->  ( B  =  B  <-> 
( -.  B R B  /\  -.  B R B ) ) )
245, 23mpbii 147 . . . . 5  |-  ( ph  ->  ( -.  B R B  /\  -.  B R B ) )
2524simpld 111 . . . 4  |-  ( ph  ->  -.  B R B )
2625adantr 272 . . 3  |-  ( (
ph  /\  x  e.  { B } )  ->  -.  B R B )
27 elsni 3492 . . . . . 6  |-  ( x  e.  { B }  ->  x  =  B )
2827breq2d 3887 . . . . 5  |-  ( x  e.  { B }  ->  ( B R x  <-> 
B R B ) )
2928notbid 633 . . . 4  |-  ( x  e.  { B }  ->  ( -.  B R x  <->  -.  B R B ) )
3029adantl 273 . . 3  |-  ( (
ph  /\  x  e.  { B } )  -> 
( -.  B R x  <->  -.  B R B ) )
3126, 30mpbird 166 . 2  |-  ( (
ph  /\  x  e.  { B } )  ->  -.  B R x )
321, 2, 4, 31supmaxti 6806 1  |-  ( ph  ->  sup ( { B } ,  A ,  R )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1299    e. wcel 1448   A.wral 2375   {csn 3474   class class class wbr 3875   supcsup 6784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-iota 5024  df-riota 5662  df-sup 6786
This theorem is referenced by:  infsnti  6832
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