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Theorem oneluni 4628
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 4624 . 2  |-  ( B  e.  A  ->  B  C_  A )
3 ssequn2 3457 . 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 1649    e. wcel 1717    u. cun 3255    C_ wss 3257   Oncon0 4516
This theorem is referenced by:  onun2i  4631
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ral 2648  df-rex 2649  df-v 2895  df-un 3262  df-in 3264  df-ss 3271  df-uni 3952  df-tr 4238  df-po 4438  df-so 4439  df-fr 4476  df-we 4478  df-ord 4519  df-on 4520
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