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Theorem suplub2 7207
Description: Bidirectional form of suplub 7206. (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
Dummy variable  w is distinct from all other variables.
Allowed substitution hints:    ph( x, y, z)    C( x, y)

Proof of Theorem suplub2
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 7206 . . 3  |-  ( ph  ->  ( ( C  e.  A  /\  C R sup ( B ,  A ,  R )
)  ->  E. z  e.  B  C R
z ) )
43expdimp 428 . 2  |-  ( (
ph  /\  C  e.  A )  ->  ( C R sup ( B ,  A ,  R
)  ->  E. z  e.  B  C R
z ) )
5 breq2 4028 . . . 4  |-  ( z  =  w  ->  ( C R z  <->  C R w ) )
65cbvrexv 2766 . . 3  |-  ( E. z  e.  B  C R z  <->  E. w  e.  B  C R w )
7 breq2 4028 . . . . . . 7  |-  ( sup ( B ,  A ,  R )  =  w  ->  ( C R sup ( B ,  A ,  R )  <->  C R w ) )
87biimprd 216 . . . . . 6  |-  ( sup ( B ,  A ,  R )  =  w  ->  ( C R w  ->  C R sup ( B ,  A ,  R ) ) )
98a1i 12 . . . . 5  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  ( sup ( B ,  A ,  R )  =  w  ->  ( C R w  ->  C R sup ( B ,  A ,  R ) ) ) )
101ad2antrr 708 . . . . . . 7  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  R  Or  A )
11 simplr 733 . . . . . . 7  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  C  e.  A )
12 suplub2.3 . . . . . . . . 9  |-  ( ph  ->  B  C_  A )
1312adantr 453 . . . . . . . 8  |-  ( (
ph  /\  C  e.  A )  ->  B  C_  A )
1413sselda 3181 . . . . . . 7  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  w  e.  A )
151, 2supcl 7204 . . . . . . . 8  |-  ( ph  ->  sup ( B ,  A ,  R )  e.  A )
1615ad2antrr 708 . . . . . . 7  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  sup ( B ,  A ,  R )  e.  A
)
17 sotr 4335 . . . . . . 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 1364 . . . . 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 7205 . . . . . . . 8  |-  ( ph  ->  ( w  e.  B  ->  -.  sup ( B ,  A ,  R
) R w ) )
2120adantr 453 . . . . . . 7  |-  ( (
ph  /\  C  e.  A )  ->  (
w  e.  B  ->  -.  sup ( B ,  A ,  R ) R w ) )
2221imp 420 . . . . . 6  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  -.  sup ( B ,  A ,  R ) R w )
23 sotric 4339 . . . . . . . 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 321 . . . . . 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 225 . . . . 5  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  ( sup ( B ,  A ,  R )  =  w  \/  w R sup ( B ,  A ,  R ) ) )
279, 19, 26mpjaod 372 . . . 4  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  ( C R w  ->  C R sup ( B ,  A ,  R )
) )
2827rexlimdva 2668 . . 3  |-  ( (
ph  /\  C  e.  A )  ->  ( E. w  e.  B  C R w  ->  C R sup ( B ,  A ,  R )
) )
296, 28syl5bi 210 . 2  |-  ( (
ph  /\  C  e.  A )  ->  ( E. z  e.  B  C R z  ->  C R sup ( B ,  A ,  R )
) )
304, 29impbid 185 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 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1624    e. wcel 1685   A.wral 2544   E.wrex 2545    C_ wss 3153   class class class wbr 4024    Or wor 4312   supcsup 7188
This theorem is referenced by:  suprlub  9711  infmrgelb  9729  supxrlub  10638  infmxrgelb  10647  infrglb  27121
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-po 4313  df-so 4314  df-iota 6252  df-riota 6299  df-sup 7189
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