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Theorem oneluni 4282
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 4280 . 2  |-  ( B  e.  A  ->  B  C_  A )
3 ssequn2 3188 . 2  |-  ( B 
C_  A  <->  ( A  u.  B )  =  A )
42, 3sylib 121 1  |-  ( B  e.  A  ->  ( A  u.  B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1296    e. wcel 1445    u. cun 3011    C_ wss 3013   Oncon0 4214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-uni 3676  df-tr 3959  df-iord 4217  df-on 4219
This theorem is referenced by: (None)
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