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Theorem suplub2 7166
Description: Bidirectional form of suplub 7165. (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
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 7165 . . 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 3987 . . . 4  |-  ( z  =  w  ->  ( C R z  <->  C R w ) )
65cbvrexv 2735 . . 3  |-  ( E. z  e.  B  C R z  <->  E. w  e.  B  C R w )
7 breq2 3987 . . . . . . 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 709 . . . . . . 7  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  R  Or  A )
11 simplr 734 . . . . . . 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 3141 . . . . . . 7  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  w  e.  A )
151, 2supcl 7163 . . . . . . . 8  |-  ( ph  ->  sup ( B ,  A ,  R )  e.  A )
1615ad2antrr 709 . . . . . . 7  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  sup ( B ,  A ,  R )  e.  A
)
17 sotr 4294 . . . . . . 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 1189 . . . . . 6  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  (
( C R w  /\  w R sup ( B ,  A ,  R ) )  ->  C R sup ( B ,  A ,  R
) ) )
1918exp3acom23 1368 . . . . 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 7164 . . . . . . . 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 4298 . . . . . . . 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 1185 . . . . . . 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 2640 . . 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 1619    e. wcel 1621   A.wral 2516   E.wrex 2517    C_ wss 3113   class class class wbr 3983    Or wor 4271   supcsup 7147
This theorem is referenced by:  suprlub  9670  infmrgelb  9688  supxrlub  10596  infmxrgelb  10605
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-po 4272  df-so 4273  df-iota 6211  df-riota 6258  df-sup 7148
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