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Theorem supnub 7467
 Description: An upper bound is not less than the supremum. (Contributed by NM, 13-Oct-2004.)
Hypotheses
Ref Expression
supmo.1
supcl.2
Assertion
Ref Expression
supnub
Distinct variable groups:   ,,,   ,,,   ,,,   ,
Allowed substitution hints:   (,,)   (,)

Proof of Theorem supnub
StepHypRef Expression
1 supmo.1 . . . . . 6
2 supcl.2 . . . . . 6
31, 2suplub 7465 . . . . 5
43expdimp 427 . . . 4
5 dfrex2 2718 . . . 4
64, 5syl6ib 218 . . 3
76con2d 109 . 2
87expimpd 587 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   wcel 1725  wral 2705  wrex 2706   class class class wbr 4212   wor 4502  csup 7445 This theorem is referenced by:  supmax  7470  dgrlb  20155  supssd  24098 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-po 4503  df-so 4504  df-iota 5418  df-riota 6549  df-sup 7446
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