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Theorem eqinftid 7017
Description: Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
Hypotheses
Ref Expression
eqinfti.ti  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
eqinftid.2  |-  ( ph  ->  C  e.  A )
eqinftid.3  |-  ( (
ph  /\  y  e.  B )  ->  -.  y R C )
eqinftid.4  |-  ( (
ph  /\  ( y  e.  A  /\  C R y ) )  ->  E. z  e.  B  z R y )
Assertion
Ref Expression
eqinftid  |-  ( ph  -> inf ( B ,  A ,  R )  =  C )
Distinct variable groups:    u, A, v, y, z    ph, u, v    u, R, v, y, z    u, B, v, y, z    u, C, v, y, z    ph, y
Allowed substitution hint:    ph( z)

Proof of Theorem eqinftid
StepHypRef Expression
1 eqinftid.2 . 2  |-  ( ph  ->  C  e.  A )
2 eqinftid.3 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  -.  y R C )
32ralrimiva 2550 . 2  |-  ( ph  ->  A. y  e.  B  -.  y R C )
4 eqinftid.4 . . . 4  |-  ( (
ph  /\  ( y  e.  A  /\  C R y ) )  ->  E. z  e.  B  z R y )
54expr 375 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( C R y  ->  E. z  e.  B  z R
y ) )
65ralrimiva 2550 . 2  |-  ( ph  ->  A. y  e.  A  ( C R y  ->  E. z  e.  B  z R y ) )
7 eqinfti.ti . . 3  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
87eqinfti 7016 . 2  |-  ( ph  ->  ( ( C  e.  A  /\  A. y  e.  B  -.  y R C  /\  A. y  e.  A  ( C R y  ->  E. z  e.  B  z R
y ) )  -> inf ( B ,  A ,  R )  =  C ) )
91, 3, 6, 8mp3and 1340 1  |-  ( ph  -> inf ( B ,  A ,  R )  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   class class class wbr 4002  infcinf 6979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-cnv 4633  df-iota 5177  df-riota 5828  df-sup 6980  df-inf 6981
This theorem is referenced by:  infminti  7023
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