Theorem onun2i 6401

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 6387	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)
4	1, 2, 3	mp2an 690	1 ⊢ (𝐴 ∪ 𝐵) ∈ On

Syntax hints:   ∈ wcel 2104   ∪ cun 3890  Oncon0 6281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-tr 5199  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-ord 6284  df-on 6285

This theorem is referenced by:  rankunb  9652  rankelun  9674  rankelpr  9675  inar1  10577