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Theorem suplub2 7212
Description: Bidirectional form of suplub 7211. (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 7211 . . 3  |-  ( ph  ->  ( ( C  e.  A  /\  C R sup ( B ,  A ,  R )
)  ->  E. z  e.  B  C R
z ) )
43expdimp 426 . 2  |-  ( (
ph  /\  C  e.  A )  ->  ( C R sup ( B ,  A ,  R
)  ->  E. z  e.  B  C R
z ) )
5 breq2 4027 . . . 4  |-  ( z  =  w  ->  ( C R z  <->  C R w ) )
65cbvrexv 2765 . . 3  |-  ( E. z  e.  B  C R z  <->  E. w  e.  B  C R w )
7 breq2 4027 . . . . . . 7  |-  ( sup ( B ,  A ,  R )  =  w  ->  ( C R sup ( B ,  A ,  R )  <->  C R w ) )
87biimprd 214 . . . . . 6  |-  ( sup ( B ,  A ,  R )  =  w  ->  ( C R w  ->  C R sup ( B ,  A ,  R ) ) )
98a1i 10 . . . . 5  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  ( sup ( B ,  A ,  R )  =  w  ->  ( C R w  ->  C R sup ( B ,  A ,  R ) ) ) )
101ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  R  Or  A )
11 simplr 731 . . . . . . 7  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  C  e.  A )
12 suplub2.3 . . . . . . . . 9  |-  ( ph  ->  B  C_  A )
1312adantr 451 . . . . . . . 8  |-  ( (
ph  /\  C  e.  A )  ->  B  C_  A )
1413sselda 3180 . . . . . . 7  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  w  e.  A )
151, 2supcl 7209 . . . . . . . 8  |-  ( ph  ->  sup ( B ,  A ,  R )  e.  A )
1615ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  sup ( B ,  A ,  R )  e.  A
)
17 sotr 4336 . . . . . . 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 1184 . . . . . 6  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  (
( C R w  /\  w R sup ( B ,  A ,  R ) )  ->  C R sup ( B ,  A ,  R
) ) )
1918exp3acom23 1362 . . . . 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 7210 . . . . . . . 8  |-  ( ph  ->  ( w  e.  B  ->  -.  sup ( B ,  A ,  R
) R w ) )
2120adantr 451 . . . . . . 7  |-  ( (
ph  /\  C  e.  A )  ->  (
w  e.  B  ->  -.  sup ( B ,  A ,  R ) R w ) )
2221imp 418 . . . . . 6  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  -.  sup ( B ,  A ,  R ) R w )
23 sotric 4340 . . . . . . . 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 1180 . . . . . . 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 319 . . . . . 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 223 . . . . 5  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  ( sup ( B ,  A ,  R )  =  w  \/  w R sup ( B ,  A ,  R ) ) )
279, 19, 26mpjaod 370 . . . 4  |-  ( ( ( ph  /\  C  e.  A )  /\  w  e.  B )  ->  ( C R w  ->  C R sup ( B ,  A ,  R )
) )
2827rexlimdva 2667 . . 3  |-  ( (
ph  /\  C  e.  A )  ->  ( E. w  e.  B  C R w  ->  C R sup ( B ,  A ,  R )
) )
296, 28syl5bi 208 . 2  |-  ( (
ph  /\  C  e.  A )  ->  ( E. z  e.  B  C R z  ->  C R sup ( B ,  A ,  R )
) )
304, 29impbid 183 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 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023    Or wor 4313   supcsup 7193
This theorem is referenced by:  suprlub  9716  infmrgelb  9734  supxrlub  10644  infmxrgelb  10653  infrglb  27722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-po 4314  df-so 4315  df-iota 5219  df-riota 6304  df-sup 7194
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