Theorem onun2i 4450

Description: The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.) (Constructive proof by Jim Kingdon, 25-Jul-2019.)

Hypotheses

Ref	Expression
onun2i.1	⊢ 𝐴 ∈ On
onun2i.2	⊢ 𝐵 ∈ On

Assertion

Ref	Expression
onun2i	⊢ (𝐴 ∪ 𝐵) ∈ On

Proof of Theorem onun2i

Step	Hyp	Ref	Expression
1		onun2i.1	. 2 ⊢ 𝐴 ∈ On
2		onun2i.2	. 2 ⊢ 𝐵 ∈ On
3		onun2 4449	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)
4	1, 2, 3	mp2an 423	1 ⊢ (𝐴 ∪ 𝐵) ∈ On

Syntax hints:   ∈ wcel 2128   ∪ cun 3100  Oncon0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-uni 3773  df-tr 4063  df-iord 4326  df-on 4328

This theorem is referenced by: (None)