ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  infminti Unicode version

Theorem infminti 6914
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 521 . . 3  |-  ( (
ph  /\  ( y  e.  A  /\  C R y ) )  ->  C R y )
6 breq1 3932 . . . 4  |-  ( z  =  C  ->  (
z R y  <->  C R
y ) )
76rspcev 2789 . . 3  |-  ( ( C  e.  B  /\  C R y )  ->  E. z  e.  B  z R y )
84, 5, 7syl2an2r 584 . 2  |-  ( (
ph  /\  ( y  e.  A  /\  C R y ) )  ->  E. z  e.  B  z R y )
91, 2, 3, 8eqinftid 6908 1  |-  ( ph  -> inf ( B ,  A ,  R )  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   E.wrex 2417   class class class wbr 3929  infcinf 6870
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 4046  ax-pow 4098  ax-pr 4131
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 3737  df-br 3930  df-opab 3990  df-cnv 4547  df-iota 5088  df-riota 5730  df-sup 6871  df-inf 6872
This theorem is referenced by:  lbinf  8718  lcmgcdlem  11769  pilem3  12886  inffz  13329  taupi  13330
  Copyright terms: Public domain W3C validator