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Theorem infminti 7000
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 527 . . 3  |-  ( (
ph  /\  ( y  e.  A  /\  C R y ) )  ->  C R y )
6 breq1 3990 . . . 4  |-  ( z  =  C  ->  (
z R y  <->  C R
y ) )
76rspcev 2834 . . 3  |-  ( ( C  e.  B  /\  C R y )  ->  E. z  e.  B  z R y )
84, 5, 7syl2an2r 590 . 2  |-  ( (
ph  /\  ( y  e.  A  /\  C R y ) )  ->  E. z  e.  B  z R y )
91, 2, 3, 8eqinftid 6994 1  |-  ( ph  -> inf ( B ,  A ,  R )  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   E.wrex 2449   class class class wbr 3987  infcinf 6956
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-cnv 4617  df-iota 5158  df-riota 5806  df-sup 6957  df-inf 6958
This theorem is referenced by:  lbinf  8851  lcmgcdlem  12018  pilem3  13457  inffz  14061  taupi  14062
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