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Theorem supnub 7717
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 7715 . . . . 5  |-  ( ph  ->  ( ( C  e.  A  /\  C R sup ( B ,  A ,  R )
)  ->  E. z  e.  B  C R
z ) )
43expdimp 437 . . . 4  |-  ( (
ph  /\  C  e.  A )  ->  ( C R sup ( B ,  A ,  R
)  ->  E. z  e.  B  C R
z ) )
5 dfrex2 2733 . . . 4  |-  ( E. z  e.  B  C R z  <->  -.  A. z  e.  B  -.  C R z )
64, 5syl6ib 226 . . 3  |-  ( (
ph  /\  C  e.  A )  ->  ( C R sup ( B ,  A ,  R
)  ->  -.  A. z  e.  B  -.  C R z ) )
76con2d 115 . 2  |-  ( (
ph  /\  C  e.  A )  ->  ( A. z  e.  B  -.  C R z  ->  -.  C R sup ( B ,  A ,  R ) ) )
87expimpd 603 1  |-  ( ph  ->  ( ( C  e.  A  /\  A. z  e.  B  -.  C R z )  ->  -.  C R sup ( B ,  A ,  R ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    e. wcel 1756   A.wral 2720   E.wrex 2721   class class class wbr 4297    Or wor 4645   supcsup 7695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-po 4646  df-so 4647  df-iota 5386  df-riota 6057  df-sup 7696
This theorem is referenced by:  supmax  7720  dgrlb  21709  supssd  26009
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