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Theorem suplub2 7400
Description: Bidirectional form of suplub 7399. (Contributed by Mario Carneiro, 6-Sep-2014.)
Hypotheses
Ref Expression
supmo.1  |-  ( ph  ->  R  Or  A )
supcl.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 ) ) )
suplub2.3  |-  ( ph  ->  B  C_  A )
Assertion
Ref Expression
suplub2  |-  ( (
ph  /\  C  e.  A )  ->  ( C R sup ( B ,  A ,  R
)  <->  E. z  e.  B  C R z ) )
Distinct variable groups:    x, y,
z, A    x, R, y, z    x, B, y, z    z, C
Allowed substitution hints:    ph( x, y, z)    C( x, y)

Proof of Theorem suplub2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 supmo.1 . . . 4  |-  ( ph  ->  R  Or  A )
2 supcl.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, 2suplub 7399 . . 3  |-  ( ph  ->  ( ( C  e.  A  /\  C R sup ( B ,  A ,  R )
)  ->  E. z  e.  B  C R
z ) )
43expdimp 427 . 2  |-  ( (
ph  /\  C  e.  A )  ->  ( C R sup ( B ,  A ,  R
)  ->  E. z  e.  B  C R
z ) )
5 breq2 4158 . . . 4  |-  ( z  =  w  ->  ( C R z  <->  C R w ) )
65cbvrexv 2877 . . 3  |-  ( E. z  e.  B  C R z  <->  E. w  e.  B  C R w )
7 breq2 4158 . . . . . . 7  |-  ( sup ( B ,  A ,  R )  =  w  ->  ( C R sup ( B ,  A ,  R )  <->  C R w ) )
87biimprd 215 . . . . . 6  |-  ( sup ( B ,  A ,  R )  =  w  ->  ( C R w  ->  C R sup ( B ,  A ,  R ) ) )
98a1i 11 . . . . 5  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  ( sup ( B ,  A ,  R )  =  w  ->  ( C R w  ->  C R sup ( B ,  A ,  R ) ) ) )
101ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  R  Or  A )
11 simplr 732 . . . . . . 7  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  C  e.  A )
12 suplub2.3 . . . . . . . . 9  |-  ( ph  ->  B  C_  A )
1312adantr 452 . . . . . . . 8  |-  ( (
ph  /\  C  e.  A )  ->  B  C_  A )
1413sselda 3292 . . . . . . 7  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  w  e.  A )
151, 2supcl 7397 . . . . . . . 8  |-  ( ph  ->  sup ( B ,  A ,  R )  e.  A )
1615ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  sup ( B ,  A ,  R )  e.  A
)
17 sotr 4467 . . . . . . 7  |-  ( ( R  Or  A  /\  ( C  e.  A  /\  w  e.  A  /\  sup ( B ,  A ,  R )  e.  A ) )  -> 
( ( C R w  /\  w R sup ( B ,  A ,  R )
)  ->  C R sup ( B ,  A ,  R ) ) )
1810, 11, 14, 16, 17syl13anc 1186 . . . . . 6  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  (
( C R w  /\  w R sup ( B ,  A ,  R ) )  ->  C R sup ( B ,  A ,  R
) ) )
1918exp3acom23 1378 . . . . 5  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  (
w R sup ( B ,  A ,  R )  ->  ( C R w  ->  C R sup ( B ,  A ,  R )
) ) )
201, 2supub 7398 . . . . . . . 8  |-  ( ph  ->  ( w  e.  B  ->  -.  sup ( B ,  A ,  R
) R w ) )
2120adantr 452 . . . . . . 7  |-  ( (
ph  /\  C  e.  A )  ->  (
w  e.  B  ->  -.  sup ( B ,  A ,  R ) R w ) )
2221imp 419 . . . . . 6  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  -.  sup ( B ,  A ,  R ) R w )
23 sotric 4471 . . . . . . . 8  |-  ( ( R  Or  A  /\  ( sup ( B ,  A ,  R )  e.  A  /\  w  e.  A ) )  -> 
( sup ( B ,  A ,  R
) R w  <->  -.  ( sup ( B ,  A ,  R )  =  w  \/  w R sup ( B ,  A ,  R ) ) ) )
2410, 16, 14, 23syl12anc 1182 . . . . . . 7  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  ( sup ( B ,  A ,  R ) R w  <->  -.  ( sup ( B ,  A ,  R
)  =  w  \/  w R sup ( B ,  A ,  R ) ) ) )
2524con2bid 320 . . . . . 6  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  (
( sup ( B ,  A ,  R
)  =  w  \/  w R sup ( B ,  A ,  R ) )  <->  -.  sup ( B ,  A ,  R ) R w ) )
2622, 25mpbird 224 . . . . 5  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  ( sup ( B ,  A ,  R )  =  w  \/  w R sup ( B ,  A ,  R ) ) )
279, 19, 26mpjaod 371 . . . 4  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  ( C R w  ->  C R sup ( B ,  A ,  R )
) )
2827rexlimdva 2774 . . 3  |-  ( (
ph  /\  C  e.  A )  ->  ( E. w  e.  B  C R w  ->  C R sup ( B ,  A ,  R )
) )
296, 28syl5bi 209 . 2  |-  ( (
ph  /\  C  e.  A )  ->  ( E. z  e.  B  C R z  ->  C R sup ( B ,  A ,  R )
) )
304, 29impbid 184 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    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650   E.wrex 2651    C_ wss 3264   class class class wbr 4154    Or wor 4444   supcsup 7381
This theorem is referenced by:  suprlub  9903  infmrgelb  9921  supxrlub  10837  infmxrgelb  10846  infrglb  27391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-po 4445  df-so 4446  df-iota 5359  df-riota 6486  df-sup 7382
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