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Theorem supcl 7463
 Description: A supremum belongs to its base class (closure law). See also supub 7464 and suplub 7465. (Contributed by NM, 12-Oct-2004.)
Hypotheses
Ref Expression
supmo.1
supcl.2
Assertion
Ref Expression
supcl
Distinct variable groups:   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem supcl
StepHypRef Expression
1 supmo.1 . . 3
2 supcl.2 . . 3
31, 2supval2 7460 . 2
41, 2supeu 7459 . . 3
5 riotacl 6564 . . 3
64, 5syl 16 . 2
73, 6eqeltrd 2510 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   wcel 1725  wral 2705  wrex 2706  wreu 2707   class class class wbr 4212   wor 4502  crio 6542  csup 7445 This theorem is referenced by:  suplub2  7466  supmax  7470  supiso  7477  suprcl  9968  infmsup  9986  supxrcl  10893  infmxrcl  10895  dgrcl  20152  supssd  24098  xrsupssd  24125  wzel  25575  wsuccl  25578  supclt  26440 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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