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Theorem suplub2ti 6509
Description: Bidirectional form of suplubti 6508. (Contributed by Jim Kingdon, 17-Jan-2022.)
Hypotheses
Ref Expression
supmoti.ti  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
supclti.2  |-  ( 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 ) ) )
suplub2ti.or  |-  ( ph  ->  R  Or  A )
suplub2ti.3  |-  ( ph  ->  B  C_  A )
Assertion
Ref Expression
suplub2ti  |-  ( (
ph  /\  C  e.  A )  ->  ( C R sup ( B ,  A ,  R
)  <->  E. z  e.  B  C R z ) )
Distinct variable groups:    u, A, v, x    y, A, x, z    x, B, y, z    u, R, v, x    y, R, z    ph, u, v, x    z, C
Allowed substitution hints:    ph( y, z)    B( v, u)    C( x, y, v, u)

Proof of Theorem suplub2ti
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 supmoti.ti . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
2 supclti.2 . . . 4  |-  ( 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 ) ) )
31, 2suplubti 6508 . . 3  |-  ( ph  ->  ( ( C  e.  A  /\  C R sup ( B ,  A ,  R )
)  ->  E. z  e.  B  C R
z ) )
43expdimp 255 . 2  |-  ( (
ph  /\  C  e.  A )  ->  ( C R sup ( B ,  A ,  R
)  ->  E. z  e.  B  C R
z ) )
5 breq2 3810 . . . 4  |-  ( z  =  w  ->  ( C R z  <->  C R w ) )
65cbvrexv 2583 . . 3  |-  ( E. z  e.  B  C R z  <->  E. w  e.  B  C R w )
7 simplll 500 . . . . . . 7  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  ph )
8 simplr 497 . . . . . . 7  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  w  e.  B )
91, 2supubti 6507 . . . . . . 7  |-  ( ph  ->  ( w  e.  B  ->  -.  sup ( B ,  A ,  R
) R w ) )
107, 8, 9sylc 61 . . . . . 6  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  -.  sup ( B ,  A ,  R ) R w )
11 simpr 108 . . . . . . 7  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  C R w )
12 suplub2ti.or . . . . . . . . 9  |-  ( ph  ->  R  Or  A )
1312ad3antrrr 476 . . . . . . . 8  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  R  Or  A )
14 simpllr 501 . . . . . . . 8  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  C  e.  A )
15 suplub2ti.3 . . . . . . . . . 10  |-  ( ph  ->  B  C_  A )
1615ad3antrrr 476 . . . . . . . . 9  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  B  C_  A )
1716, 8sseldd 3010 . . . . . . . 8  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  w  e.  A )
181, 2supclti 6506 . . . . . . . . 9  |-  ( ph  ->  sup ( B ,  A ,  R )  e.  A )
1918ad3antrrr 476 . . . . . . . 8  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  sup ( B ,  A ,  R )  e.  A
)
20 sowlin 4104 . . . . . . . 8  |-  ( ( R  Or  A  /\  ( C  e.  A  /\  w  e.  A  /\  sup ( B ,  A ,  R )  e.  A ) )  -> 
( C R w  ->  ( C R sup ( B ,  A ,  R )  \/  sup ( B ,  A ,  R ) R w ) ) )
2113, 14, 17, 19, 20syl13anc 1172 . . . . . . 7  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  ( C R w  ->  ( C R sup ( B ,  A ,  R
)  \/  sup ( B ,  A ,  R ) R w ) ) )
2211, 21mpd 13 . . . . . 6  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  ( C R sup ( B ,  A ,  R
)  \/  sup ( B ,  A ,  R ) R w ) )
2310, 22ecased 1281 . . . . 5  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  C R sup ( B ,  A ,  R )
)
2423ex 113 . . . 4  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  ( C R w  ->  C R sup ( B ,  A ,  R )
) )
2524rexlimdva 2482 . . 3  |-  ( (
ph  /\  C  e.  A )  ->  ( E. w  e.  B  C R w  ->  C R sup ( B ,  A ,  R )
) )
266, 25syl5bi 150 . 2  |-  ( (
ph  /\  C  e.  A )  ->  ( E. z  e.  B  C R z  ->  C R sup ( B ,  A ,  R )
) )
274, 26impbid 127 1  |-  ( (
ph  /\  C  e.  A )  ->  ( C R sup ( B ,  A ,  R
)  <->  E. z  e.  B  C R z ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    e. wcel 1434   A.wral 2353   E.wrex 2354    C_ wss 2983   class class class wbr 3806    Or wor 4079   supcsup 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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
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 2826  df-un 2987  df-in 2989  df-ss 2996  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-iso 4081  df-iota 4918  df-riota 5520  df-sup 6492
This theorem is referenced by:  suprlubex  8133
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