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Theorem infsnti 6920
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 6875 . 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 6909 . . 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 6895 . 2  |-  ( ph  ->  sup ( { B } ,  A ,  `' R )  =  B )
61, 5syl5eq 2184 1  |-  ( ph  -> inf ( { B } ,  A ,  R )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   {csn 3527   class class class wbr 3932   `'ccnv 4541   supcsup 6872  infcinf 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-br 3933  df-opab 3993  df-cnv 4550  df-iota 5091  df-riota 5733  df-sup 6874  df-inf 6875
This theorem is referenced by: (None)
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