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Theorem onun2i 6436
Description: The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.)
Hypotheses
Ref Expression
on.1 𝐴 ∈ On
on.2 𝐵 ∈ On
Assertion
Ref Expression
onun2i (𝐴𝐵) ∈ On

Proof of Theorem onun2i
StepHypRef Expression
1 on.1 . 2 𝐴 ∈ On
2 on.2 . 2 𝐵 ∈ On
3 onun2 6422 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
41, 2, 3mp2an 690 1 (𝐴𝐵) ∈ On
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  cun 3906  Oncon0 6315
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-tr 5221  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-ord 6318  df-on 6319
This theorem is referenced by:  rankunb  9744  rankelun  9766  rankelpr  9767  inar1  10669  addsprop  34291  negsprop  34328
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