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Theorem suplub2ti 6888
Description: Bidirectional form of suplubti 6887. (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 6887 . . 3  |-  ( ph  ->  ( ( C  e.  A  /\  C R sup ( B ,  A ,  R )
)  ->  E. z  e.  B  C R
z ) )
43expdimp 257 . 2  |-  ( (
ph  /\  C  e.  A )  ->  ( C R sup ( B ,  A ,  R
)  ->  E. z  e.  B  C R
z ) )
5 breq2 3933 . . . 4  |-  ( z  =  w  ->  ( C R z  <->  C R w ) )
65cbvrexv 2655 . . 3  |-  ( E. z  e.  B  C R z  <->  E. w  e.  B  C R w )
7 simplll 522 . . . . . . 7  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  ph )
8 simplr 519 . . . . . . 7  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  w  e.  B )
91, 2supubti 6886 . . . . . . 7  |-  ( ph  ->  ( w  e.  B  ->  -.  sup ( B ,  A ,  R
) R w ) )
107, 8, 9sylc 62 . . . . . 6  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  -.  sup ( B ,  A ,  R ) R w )
11 simpr 109 . . . . . . 7  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  C R w )
12 suplub2ti.or . . . . . . . . 9  |-  ( ph  ->  R  Or  A )
1312ad3antrrr 483 . . . . . . . 8  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  R  Or  A )
14 simpllr 523 . . . . . . . 8  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  C  e.  A )
15 suplub2ti.3 . . . . . . . . . 10  |-  ( ph  ->  B  C_  A )
1615ad3antrrr 483 . . . . . . . . 9  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  B  C_  A )
1716, 8sseldd 3098 . . . . . . . 8  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  w  e.  A )
181, 2supclti 6885 . . . . . . . . 9  |-  ( ph  ->  sup ( B ,  A ,  R )  e.  A )
1918ad3antrrr 483 . . . . . . . 8  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  sup ( B ,  A ,  R )  e.  A
)
20 sowlin 4242 . . . . . . . 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 1218 . . . . . . 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 1327 . . . . 5  |-  ( ( ( ( ph  /\  C  e.  A )  /\  w  e.  B
)  /\  C R w )  ->  C R sup ( B ,  A ,  R )
)
2423ex 114 . . . 4  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  ( C R w  ->  C R sup ( B ,  A ,  R )
) )
2524rexlimdva 2549 . . 3  |-  ( (
ph  /\  C  e.  A )  ->  ( E. w  e.  B  C R w  ->  C R sup ( B ,  A ,  R )
) )
266, 25syl5bi 151 . 2  |-  ( (
ph  /\  C  e.  A )  ->  ( E. z  e.  B  C R z  ->  C R sup ( B ,  A ,  R )
) )
274, 26impbid 128 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 103    <-> wb 104    \/ wo 697    e. wcel 1480   A.wral 2416   E.wrex 2417    C_ wss 3071   class class class wbr 3929    Or wor 4217   supcsup 6869
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-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
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-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iso 4219  df-iota 5088  df-riota 5730  df-sup 6871
This theorem is referenced by:  suprlubex  8722
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