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Theorem inflbti 6911
Description: An infimum is a lower bound. See also infclti 6910 and infglbti 6912. (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 6906 . . . . 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 6905 . . . . 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 6886 . . . 4  |-  ( ph  ->  ( C  e.  B  ->  -.  sup ( B ,  A ,  `' R ) `' R C ) )
65imp 123 . . 3  |-  ( (
ph  /\  C  e.  B )  ->  -.  sup ( B ,  A ,  `' R ) `' R C )
7 df-inf 6872 . . . . . 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 3941 . . . 4  |-  ( (
ph  /\  C  e.  B )  ->  ( C Rinf ( B ,  A ,  R )  <->  C R sup ( B ,  A ,  `' R ) ) )
102, 4supclti 6885 . . . . 5  |-  ( ph  ->  sup ( B ,  A ,  `' R
)  e.  A )
11 brcnvg 4720 . . . . . 6  |-  ( ( sup ( B ,  A ,  `' R
)  e.  A  /\  C  e.  B )  ->  ( sup ( B ,  A ,  `' R ) `' R C 
<->  C R sup ( B ,  A ,  `' R ) ) )
1211bicomd 140 . . . . 5  |-  ( ( sup ( B ,  A ,  `' R
)  e.  A  /\  C  e.  B )  ->  ( C R sup ( B ,  A ,  `' R )  <->  sup ( B ,  A ,  `' R ) `' R C ) )
1310, 12sylan 281 . . . 4  |-  ( (
ph  /\  C  e.  B )  ->  ( C R sup ( B ,  A ,  `' R )  <->  sup ( B ,  A ,  `' R ) `' R C ) )
149, 13bitrd 187 . . 3  |-  ( (
ph  /\  C  e.  B )  ->  ( C Rinf ( B ,  A ,  R )  <->  sup ( B ,  A ,  `' R ) `' R C ) )
156, 14mtbird 662 . 2  |-  ( (
ph  /\  C  e.  B )  ->  -.  C Rinf ( B ,  A ,  R )
)
1615ex 114 1  |-  ( ph  ->  ( C  e.  B  ->  -.  C Rinf ( B ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   A.wral 2416   E.wrex 2417   class class class wbr 3929   `'ccnv 4538   supcsup 6869  infcinf 6870
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-cnv 4547  df-iota 5088  df-riota 5730  df-sup 6871  df-inf 6872
This theorem is referenced by:  zssinfcl  11654  infssuzledc  11656
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