Theorem onun2i 6463

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	⊢ (𝐴 ∪ 𝐵) ∈ On

Proof of Theorem onun2i

Step	Hyp	Ref	Expression
1		on.1	. 2 ⊢ 𝐴 ∈ On
2		on.2	. 2 ⊢ 𝐵 ∈ On
3		onun2 6450	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)
4	1, 2, 3	mp2an 702	1 ⊢ (𝐴 ∪ 𝐵) ∈ On

Syntax hints:   ∈ wcel 2141   ∪ cun 3900  Oncon0 6340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-tr 5205  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-ord 6343  df-on 6344

This theorem is referenced by:  rankunb  9801  rankelun  9823  rankelpr  9824  inar1  10726  addsprop  28056  negsprop  28115  mulsproplem5  28200  mulsproplem6  28201  mulsproplem7  28202  mulsproplem8  28203  mulsprop  28210