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Theorem supsnti 6896
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 3557 . . 3  |-  ( B  e.  A  ->  B  e.  { B } )
42, 3syl 14 . 2  |-  ( ph  ->  B  e.  { B } )
5 eqid 2140 . . . . . 6  |-  B  =  B
61ralrimivva 2515 . . . . . . 7  |-  ( ph  ->  A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )
7 eqeq1 2147 . . . . . . . . . 10  |-  ( u  =  B  ->  (
u  =  v  <->  B  =  v ) )
8 breq1 3936 . . . . . . . . . . . 12  |-  ( u  =  B  ->  (
u R v  <->  B R
v ) )
98notbid 657 . . . . . . . . . . 11  |-  ( u  =  B  ->  ( -.  u R v  <->  -.  B R v ) )
10 breq2 3937 . . . . . . . . . . . 12  |-  ( u  =  B  ->  (
v R u  <->  v R B ) )
1110notbid 657 . . . . . . . . . . 11  |-  ( u  =  B  ->  ( -.  v R u  <->  -.  v R B ) )
129, 11anbi12d 465 . . . . . . . . . 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 2150 . . . . . . . . . 10  |-  ( v  =  B  ->  ( B  =  v  <->  B  =  B ) )
15 breq2 3937 . . . . . . . . . . . 12  |-  ( v  =  B  ->  ( B R v  <->  B R B ) )
1615notbid 657 . . . . . . . . . . 11  |-  ( v  =  B  ->  ( -.  B R v  <->  -.  B R B ) )
17 breq1 3936 . . . . . . . . . . . 12  |-  ( v  =  B  ->  (
v R B  <->  B R B ) )
1817notbid 657 . . . . . . . . . . 11  |-  ( v  =  B  ->  ( -.  v R B  <->  -.  B R B ) )
1916, 18anbi12d 465 . . . . . . . . . 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 2803 . . . . . . . 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 409 . . . . . . 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 274 . . 3  |-  ( (
ph  /\  x  e.  { B } )  ->  -.  B R B )
27 elsni 3546 . . . . . 6  |-  ( x  e.  { B }  ->  x  =  B )
2827breq2d 3945 . . . . 5  |-  ( x  e.  { B }  ->  ( B R x  <-> 
B R B ) )
2928notbid 657 . . . 4  |-  ( x  e.  { B }  ->  ( -.  B R x  <->  -.  B R B ) )
3029adantl 275 . . 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 6895 1  |-  ( ph  ->  sup ( { B } ,  A ,  R )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481   A.wral 2417   {csn 3528   class class class wbr 3933   supcsup 6873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2689  df-sbc 2911  df-un 3076  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-br 3934  df-iota 5092  df-riota 5734  df-sup 6875
This theorem is referenced by:  infsnti  6921
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