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Theorem inflbti 6826
Description: An infimum is a lower bound. See also infclti 6825 and infglbti 6827. (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 6821 . . . . 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 6820 . . . . 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 6801 . . . 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 6787 . . . . . 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 3887 . . . 4  |-  ( (
ph  /\  C  e.  B )  ->  ( C Rinf ( B ,  A ,  R )  <->  C R sup ( B ,  A ,  `' R ) ) )
102, 4supclti 6800 . . . . 5  |-  ( ph  ->  sup ( B ,  A ,  `' R
)  e.  A )
11 brcnvg 4658 . . . . . 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 279 . . . 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 639 . 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 1299    e. wcel 1448   A.wral 2375   E.wrex 2376   class class class wbr 3875   `'ccnv 4476   supcsup 6784  infcinf 6785
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 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-cnv 4485  df-iota 5024  df-riota 5662  df-sup 6786  df-inf 6787
This theorem is referenced by:  zssinfcl  11436  infssuzledc  11438
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