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Theorem infminti 6535
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 499 . . 3  |-  ( (
ph  /\  ( y  e.  A  /\  C R y ) )  ->  C R y )
6 breq1 3809 . . . 4  |-  ( z  =  C  ->  (
z R y  <->  C R
y ) )
76rspcev 2710 . . 3  |-  ( ( C  e.  B  /\  C R y )  ->  E. z  e.  B  z R y )
84, 5, 7syl2an2r 560 . 2  |-  ( (
ph  /\  ( y  e.  A  /\  C R y ) )  ->  E. z  e.  B  z R y )
91, 2, 3, 8eqinftid 6529 1  |-  ( ph  -> inf ( B ,  A ,  R )  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   E.wrex 2354   class class class wbr 3806  infcinf 6491
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2826  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-cnv 4400  df-iota 4918  df-riota 5520  df-sup 6492  df-inf 6493
This theorem is referenced by:  lbinf  8129  lcmgcdlem  10650
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