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Theorem inflbti 6531
Description: An infimum is a lower bound. See also infclti 6530 and infglbti 6532. (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
inflbti  |-  ( ph  ->  ( C  e.  B  ->  -.  C Rinf ( B ,  A ,  R ) ) )
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
Allowed substitution hints:    C( x, y, z, v, u)

Proof of Theorem inflbti
StepHypRef Expression
1 infclti.ti . . . . . 6  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
21cnvti 6526 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u `' R v  /\  -.  v `' R u ) ) )
3 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 ) ) )
43cnvinfex 6525 . . . . 5  |-  ( 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 ) ) )
52, 4supubti 6506 . . . 4  |-  ( ph  ->  ( C  e.  B  ->  -.  sup ( B ,  A ,  `' R ) `' R C ) )
65imp 122 . . 3  |-  ( (
ph  /\  C  e.  B )  ->  -.  sup ( B ,  A ,  `' R ) `' R C )
7 df-inf 6492 . . . . . 6  |- inf ( B ,  A ,  R
)  =  sup ( B ,  A ,  `' R )
87a1i 9 . . . . 5  |-  ( (
ph  /\  C  e.  B )  -> inf ( B ,  A ,  R
)  =  sup ( B ,  A ,  `' R ) )
98breq2d 3817 . . . 4  |-  ( (
ph  /\  C  e.  B )  ->  ( C Rinf ( B ,  A ,  R )  <->  C R sup ( B ,  A ,  `' R ) ) )
102, 4supclti 6505 . . . . 5  |-  ( ph  ->  sup ( B ,  A ,  `' R
)  e.  A )
11 brcnvg 4564 . . . . . 6  |-  ( ( sup ( B ,  A ,  `' R
)  e.  A  /\  C  e.  B )  ->  ( sup ( B ,  A ,  `' R ) `' R C 
<->  C R sup ( B ,  A ,  `' R ) ) )
1211bicomd 139 . . . . 5  |-  ( ( sup ( B ,  A ,  `' R
)  e.  A  /\  C  e.  B )  ->  ( C R sup ( B ,  A ,  `' R )  <->  sup ( B ,  A ,  `' R ) `' R C ) )
1310, 12sylan 277 . . . 4  |-  ( (
ph  /\  C  e.  B )  ->  ( C R sup ( B ,  A ,  `' R )  <->  sup ( B ,  A ,  `' R ) `' R C ) )
149, 13bitrd 186 . . 3  |-  ( (
ph  /\  C  e.  B )  ->  ( C Rinf ( B ,  A ,  R )  <->  sup ( B ,  A ,  `' R ) `' R C ) )
156, 14mtbird 631 . 2  |-  ( (
ph  /\  C  e.  B )  ->  -.  C Rinf ( B ,  A ,  R )
)
1615ex 113 1  |-  ( ph  ->  ( C  e.  B  ->  -.  C Rinf ( B ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2353   E.wrex 2354   class class class wbr 3805   `'ccnv 4390   supcsup 6489  infcinf 6490
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-cnv 4399  df-iota 4917  df-riota 5519  df-sup 6491  df-inf 6492
This theorem is referenced by:  zssinfcl  10551  infssuzledc  10553
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