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Theorem infsnti 6669
Description: The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.)
Hypotheses
Ref Expression
infsnti.ti  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
infsnti.b  |-  ( ph  ->  B  e.  A )
Assertion
Ref Expression
infsnti  |-  ( ph  -> inf ( { B } ,  A ,  R )  =  B )
Distinct variable groups:    u, A, v   
u, B, v    u, R, v    ph, u, v

Proof of Theorem infsnti
StepHypRef Expression
1 df-inf 6624 . 2  |- inf ( { B } ,  A ,  R )  =  sup ( { B } ,  A ,  `' R
)
2 infsnti.ti . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
32cnvti 6658 . . 3  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u `' R v  /\  -.  v `' R u ) ) )
4 infsnti.b . . 3  |-  ( ph  ->  B  e.  A )
53, 4supsnti 6644 . 2  |-  ( ph  ->  sup ( { B } ,  A ,  `' R )  =  B )
61, 5syl5eq 2129 1  |-  ( ph  -> inf ( { B } ,  A ,  R )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1287    e. wcel 1436   {csn 3431   class class class wbr 3820   `'ccnv 4410   supcsup 6621  infcinf 6622
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-cnv 4419  df-iota 4946  df-riota 5569  df-sup 6623  df-inf 6624
This theorem is referenced by: (None)
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