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Theorem oneluni 4686
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 4682 . 2  |-  ( B  e.  A  ->  B  C_  A )
3 ssequn2 3512 . 2  |-  ( B 
C_  A  <->  ( A  u.  B )  =  A )
42, 3sylib 189 1  |-  ( B  e.  A  ->  ( A  u.  B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    u. cun 3310    C_ wss 3312   Oncon0 4573
This theorem is referenced by:  onun2i  4689
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 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-un 3317  df-in 3319  df-ss 3326  df-uni 4008  df-tr 4295  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577
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