Theorem onun2

Description: The union of two ordinals is an ordinal. (Contributed by Scott Fenton, 9-Aug-2024)

	Ref	Expression
Assertion	onun2	|- ( ( A e. On /\ B e. On ) -> ( A u. B ) e. On )

Proof

Step	Hyp	Ref	Expression
1		ssequn1	|- ( A C_ B <-> ( A u. B ) = B )
2		eleq1a	|- ( B e. On -> ( ( A u. B ) = B -> ( A u. B ) e. On ) )
3	2	adantl	|- ( ( A e. On /\ B e. On ) -> ( ( A u. B ) = B -> ( A u. B ) e. On ) )
4	1 3	syl5bi	|- ( ( A e. On /\ B e. On ) -> ( A C_ B -> ( A u. B ) e. On ) )
5		ssequn2	|- ( B C_ A <-> ( A u. B ) = A )
6		eleq1a	|- ( A e. On -> ( ( A u. B ) = A -> ( A u. B ) e. On ) )
7	6	adantr	|- ( ( A e. On /\ B e. On ) -> ( ( A u. B ) = A -> ( A u. B ) e. On ) )
8	5 7	syl5bi	|- ( ( A e. On /\ B e. On ) -> ( B C_ A -> ( A u. B ) e. On ) )
9		eloni	|- ( A e. On -> Ord A )
10		eloni	|- ( B e. On -> Ord B )
11		ordtri2or2	|- ( ( Ord A /\ Ord B ) -> ( A C_ B \/ B C_ A ) )
12	9 10 11	syl2an	|- ( ( A e. On /\ B e. On ) -> ( A C_ B \/ B C_ A ) )
13	4 8 12	mpjaod	|- ( ( A e. On /\ B e. On ) -> ( A u. B ) e. On )