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Theorem supnub 7213
Description: An upper bound is not less than the supremum. (Contributed by NM, 13-Oct-2004.)
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 ) ) )
Assertion
Ref Expression
supnub  |-  ( ph  ->  ( ( C  e.  A  /\  A. z  e.  B  -.  C R z )  ->  -.  C R sup ( B ,  A ,  R ) ) )
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 supnub
StepHypRef Expression
1 supmo.1 . . . . . 6  |-  ( ph  ->  R  Or  A )
2 supcl.2 . . . . . 6  |-  ( 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 . . . . 5  |-  ( ph  ->  ( ( C  e.  A  /\  C R sup ( B ,  A ,  R )
)  ->  E. z  e.  B  C R
z ) )
43expdimp 426 . . . 4  |-  ( (
ph  /\  C  e.  A )  ->  ( C R sup ( B ,  A ,  R
)  ->  E. z  e.  B  C R
z ) )
5 dfrex2 2556 . . . 4  |-  ( E. z  e.  B  C R z  <->  -.  A. z  e.  B  -.  C R z )
64, 5syl6ib 217 . . 3  |-  ( (
ph  /\  C  e.  A )  ->  ( C R sup ( B ,  A ,  R
)  ->  -.  A. z  e.  B  -.  C R z ) )
76con2d 107 . 2  |-  ( (
ph  /\  C  e.  A )  ->  ( A. z  e.  B  -.  C R z  ->  -.  C R sup ( B ,  A ,  R ) ) )
87expimpd 586 1  |-  ( ph  ->  ( ( C  e.  A  /\  A. z  e.  B  -.  C R z )  ->  -.  C R sup ( B ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023    Or wor 4313   supcsup 7193
This theorem is referenced by:  supmax  7216  dgrlb  19618  supssd  23248
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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