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Theorem supval2 7452
 Description: Alternative expression for the supremum. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypotheses
Ref Expression
supmo.1
supeu.2
Assertion
Ref Expression
supval2
Distinct variable groups:   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem supval2
StepHypRef Expression
1 supmo.1 . . . 4
2 supeu.2 . . . 4
31, 2supeu 7451 . . 3
4 riotauni 6548 . . 3
53, 4syl 16 . 2
6 df-sup 7438 . 2
75, 6syl6reqr 2486 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   wceq 1652  wral 2697  wrex 2698  wreu 2699  crab 2701  cuni 4007   class class class wbr 4204   wor 4494  crio 6534  csup 7437 This theorem is referenced by:  eqsup  7453  supcl  7455  supub  7456  suplub  7457  fisupcl  7464  toslub  24183  tosglb  24184 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 2416 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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-po 4495  df-so 4496  df-iota 5410  df-riota 6541  df-sup 7438
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