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Theorem infminti 6882
Description: The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by Jim Kingdon, 18-Dec-2021.)
Hypotheses
Ref Expression
infminti.ti  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
infminti.2  |-  ( ph  ->  C  e.  A )
infminti.3  |-  ( ph  ->  C  e.  B )
infminti.4  |-  ( (
ph  /\  y  e.  B )  ->  -.  y R C )
Assertion
Ref Expression
infminti  |-  ( ph  -> inf ( B ,  A ,  R )  =  C )
Distinct variable groups:    u, A, v, y    u, B, v, y    u, C, v, y    u, R, v, y    ph, u, v, y

Proof of Theorem infminti
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 infminti.ti . 2  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
2 infminti.2 . 2  |-  ( ph  ->  C  e.  A )
3 infminti.4 . 2  |-  ( (
ph  /\  y  e.  B )  ->  -.  y R C )
4 infminti.3 . . 3  |-  ( ph  ->  C  e.  B )
5 simprr 506 . . 3  |-  ( (
ph  /\  ( y  e.  A  /\  C R y ) )  ->  C R y )
6 breq1 3902 . . . 4  |-  ( z  =  C  ->  (
z R y  <->  C R
y ) )
76rspcev 2763 . . 3  |-  ( ( C  e.  B  /\  C R y )  ->  E. z  e.  B  z R y )
84, 5, 7syl2an2r 569 . 2  |-  ( (
ph  /\  ( y  e.  A  /\  C R y ) )  ->  E. z  e.  B  z R y )
91, 2, 3, 8eqinftid 6876 1  |-  ( ph  -> inf ( B ,  A ,  R )  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   E.wrex 2394   class class class wbr 3899  infcinf 6838
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-cnv 4517  df-iota 5058  df-riota 5698  df-sup 6839  df-inf 6840
This theorem is referenced by:  lbinf  8674  lcmgcdlem  11685  pilem3  12791  inffz  13165  taupi  13166
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