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Theorem infclti 7021
Description: An infimum belongs to its base class (closure law). See also inflbti 7022 and infglbti 7023. (Contributed by Jim Kingdon, 17-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
infclti  |-  ( ph  -> inf ( B ,  A ,  R )  e.  A
)
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

Proof of Theorem infclti
StepHypRef Expression
1 df-inf 6983 . 2  |- inf ( B ,  A ,  R
)  =  sup ( B ,  A ,  `' R )
2 infclti.ti . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
32cnvti 7017 . . 3  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u `' R v  /\  -.  v `' R u ) ) )
4 infclti.ex . . . 4  |-  ( 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 ) ) )
54cnvinfex 7016 . . 3  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  x `' R y  /\  A. y  e.  A  ( y `' R x  ->  E. z  e.  B  y `' R z ) ) )
63, 5supclti 6996 . 2  |-  ( ph  ->  sup ( B ,  A ,  `' R
)  e.  A )
71, 6eqeltrid 2264 1  |-  ( ph  -> inf ( B ,  A ,  R )  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    e. wcel 2148   A.wral 2455   E.wrex 2456   class class class wbr 4003   `'ccnv 4625   supcsup 6980  infcinf 6981
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 4121  ax-pow 4174  ax-pr 4209
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 4004  df-opab 4065  df-cnv 4634  df-iota 5178  df-riota 5830  df-sup 6982  df-inf 6983
This theorem is referenced by:  infrenegsupex  9592  supminfex  9595  infregelbex  9596  infxrnegsupex  11266  infssuzledc  11945
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