Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  supcl Unicode version

Theorem supcl 7938
 Description: A supremum belongs to its base class (closure law). See also supub 7939 and suplub 7940. (Contributed by NM, 12-Oct-2004.)
Hypotheses
Ref Expression
supmo.1
supcl.2
Assertion
Ref Expression
supcl
Distinct variable groups:   ,,,   ,,,   ,,,

Proof of Theorem supcl
StepHypRef Expression
1 supmo.1 . . 3
21supval2 7935 . 2
3 supcl.2 . . . 4
41, 3supeu 7934 . . 3
5 riotacl 6272 . . 3
64, 5syl 16 . 2
72, 6eqeltrd 2545 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  /\wa 369  e.wcel 1818  A.wral 2807  E.wrex 2808  E!wreu 2809   class class class wbr 4452  Orwor 4804  iota_crio 6256  supcsup 7920 This theorem is referenced by:  suplub2  7941  supmaxOLD  7946  supiso  7954  suprcl  10528  infmsup  10546  supxrcl  11535  infmxrcl  11537  dgrcl  22630  supssd  27527  xrsupssd  27579  xrge0infssd  27581  oddpwdc  28293  wzel  29380  wsuccl  29383  supclt  30229 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-po 4805  df-so 4806  df-iota 5556  df-riota 6257  df-sup 7921
 Copyright terms: Public domain W3C validator