MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  supeq1 Unicode version

Theorem supeq1 7925
Description: Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
Assertion
Ref Expression
supeq1

Proof of Theorem supeq1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 3054 . . . . 5
2 rexeq 3055 . . . . . . 7
32imbi2d 316 . . . . . 6
43ralbidv 2896 . . . . 5
51, 4anbi12d 710 . . . 4
65rabbidv 3101 . . 3
76unieqd 4259 . 2
8 df-sup 7921 . 2
9 df-sup 7921 . 2
107, 8, 93eqtr4g 2523 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  /\wa 369  =wceq 1395  A.wral 2807  E.wrex 2808  {crab 2811  U.cuni 4249   class class class wbr 4452  supcsup 7920
This theorem is referenced by:  supeq1d  7926  supeq1i  7927  ramcl2lem  14527  odval  16558  submod  16589  bndth  21458  ioorval  21983  uniioombllem6  21997  mdegcl  22469
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-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-uni 4250  df-sup 7921
  Copyright terms: Public domain W3C validator