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

Theorem oneluni 5535
Description: An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)
Hypothesis
Ref Expression
on.1  |-  A  e.  On
Assertion
Ref Expression
oneluni  |-  ( B  e.  A  ->  ( A  u.  B )  =  A )

Proof of Theorem oneluni
StepHypRef Expression
1 on.1 . . 3  |-  A  e.  On
21onelssi 5531 . 2  |-  ( B  e.  A  ->  B  C_  A )
3 ssequn2 3607 . 2  |-  ( B 
C_  A  <->  ( A  u.  B )  =  A )
42, 3sylib 200 1  |-  ( B  e.  A  ->  ( A  u.  B )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1444    e. wcel 1887    u. cun 3402    C_ wss 3404   Oncon0 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rex 2743  df-v 3047  df-un 3409  df-in 3411  df-ss 3418  df-uni 4199  df-tr 4498  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-ord 5426  df-on 5427
This theorem is referenced by:  onun2i  5538
  Copyright terms: Public domain W3C validator