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Theorem oneluni 6281
Description: An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)
Hypothesis
Ref Expression
on.1 𝐴 ∈ On
Assertion
Ref Expression
oneluni (𝐵𝐴 → (𝐴𝐵) = 𝐴)

Proof of Theorem oneluni
StepHypRef Expression
1 on.1 . . 3 𝐴 ∈ On
21onelssi 6277 . 2 (𝐵𝐴𝐵𝐴)
3 ssequn2 4134 . 2 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐴)
42, 3sylib 221 1 (𝐵𝐴 → (𝐴𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  cun 3906  wss 3908  Oncon0 6169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-11 2161  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-ral 3135  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-uni 4814  df-tr 5149  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-ord 6172  df-on 6173
This theorem is referenced by:  onun2i  6284
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