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Theorem infnlbti 6863
Description: A lower bound is not greater than the infimum. (Contributed by Jim Kingdon, 18-Dec-2021.)
Hypotheses
Ref Expression
infclti.ti  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
infclti.ex  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )
Assertion
Ref Expression
infnlbti  |-  ( ph  ->  ( ( C  e.  A  /\  A. z  e.  B  -.  z R C )  ->  -. inf ( B ,  A ,  R ) R C ) )
Distinct variable groups:    u, A, v, x, y, z    u, B, v, x, y, z   
u, R, v, x, y, z    ph, u, v, x, y, z    z, C
Allowed substitution hints:    C( x, y, v, u)

Proof of Theorem infnlbti
StepHypRef Expression
1 infclti.ti . . . . . 6  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
2 infclti.ex . . . . . 6  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )
31, 2infglbti 6862 . . . . 5  |-  ( ph  ->  ( ( C  e.  A  /\ inf ( B ,  A ,  R ) R C )  ->  E. z  e.  B  z R C ) )
43expdimp 257 . . . 4  |-  ( (
ph  /\  C  e.  A )  ->  (inf ( B ,  A ,  R ) R C  ->  E. z  e.  B  z R C ) )
5 rexalim 2402 . . . 4  |-  ( E. z  e.  B  z R C  ->  -.  A. z  e.  B  -.  z R C )
64, 5syl6 33 . . 3  |-  ( (
ph  /\  C  e.  A )  ->  (inf ( B ,  A ,  R ) R C  ->  -.  A. z  e.  B  -.  z R C ) )
76con2d 596 . 2  |-  ( (
ph  /\  C  e.  A )  ->  ( A. z  e.  B  -.  z R C  ->  -. inf ( B ,  A ,  R ) R C ) )
87expimpd 358 1  |-  ( ph  ->  ( ( C  e.  A  /\  A. z  e.  B  -.  z R C )  ->  -. inf ( B ,  A ,  R ) R C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1461   A.wral 2388   E.wrex 2389   class class class wbr 3893  infcinf 6820
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-cnv 4505  df-iota 5044  df-riota 5682  df-sup 6821  df-inf 6822
This theorem is referenced by: (None)
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