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

Theorem supeu 7934
Description: A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by NM, 12-Oct-2004.)
Hypotheses
Ref Expression
supmo.1
supeu.2
Assertion
Ref Expression
supeu
Distinct variable groups:   , , ,   , , ,   , , ,

Proof of Theorem supeu
StepHypRef Expression
1 supeu.2 . 2
2 supmo.1 . . 3
32supmo 7932 . 2
4 reu5 3073 . 2
51, 3, 4sylanbrc 664 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  /\wa 369  A.wral 2807  E.wrex 2808  E!wreu 2809  E*wrmo 2810   class class class wbr 4452  Orwor 4804
This theorem is referenced by:  supval2  7935  eqsup  7936  supcl  7938  supub  7939  suplub  7940  fisup2g  7947  fisupcl  7948  xrsclat  27668
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-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  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-br 4453  df-po 4805  df-so 4806
  Copyright terms: Public domain W3C validator