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Theorem infclti 6878
Description: An infimum belongs to its base class (closure law). See also inflbti 6879 and infglbti 6880. (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 6840 . 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 6874 . . 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 6873 . . 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 6853 . 2  |-  ( ph  ->  sup ( B ,  A ,  `' R
)  e.  A )
71, 6eqeltrid 2204 1  |-  ( ph  -> inf ( B ,  A ,  R )  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1465   A.wral 2393   E.wrex 2394   class class class wbr 3899   `'ccnv 4508   supcsup 6837  infcinf 6838
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-cnv 4517  df-iota 5058  df-riota 5698  df-sup 6839  df-inf 6840
This theorem is referenced by:  infrenegsupex  9357  supminfex  9360  infxrnegsupex  11000  infssuzledc  11570
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