Theorem nndomog

Description: Cardinal ordering agrees with ordinal number ordering when the smaller number is a natural number. Compare with nndomo when both are natural numbers. (Contributed by NM, 17-Jun-1998) Generalize from nndomo . (Revised by RP, 5-Nov-2023)

	Ref	Expression
Assertion	nndomog	|- ( ( A e. _om /\ B e. On ) -> ( A ~<_ B <-> A C_ B ) )

Proof

Step	Hyp	Ref	Expression
1		php2	|- ( ( A e. _om /\ B C. A ) -> B ~< A )
2	1	ex	|- ( A e. _om -> ( B C. A -> B ~< A ) )
3		domnsym	|- ( A ~<_ B -> -. B ~< A )
4	2 3	nsyli	|- ( A e. _om -> ( A ~<_ B -> -. B C. A ) )
5	4	adantr	|- ( ( A e. _om /\ B e. On ) -> ( A ~<_ B -> -. B C. A ) )
6		nnord	|- ( A e. _om -> Ord A )
7		eloni	|- ( B e. On -> Ord B )
8		ordtri1	|- ( ( Ord A /\ Ord B ) -> ( A C_ B <-> -. B e. A ) )
9		ordelpss	|- ( ( Ord B /\ Ord A ) -> ( B e. A <-> B C. A ) )
10	9	ancoms	|- ( ( Ord A /\ Ord B ) -> ( B e. A <-> B C. A ) )
11	10	notbid	|- ( ( Ord A /\ Ord B ) -> ( -. B e. A <-> -. B C. A ) )
12	8 11	bitrd	|- ( ( Ord A /\ Ord B ) -> ( A C_ B <-> -. B C. A ) )
13	6 7 12	syl2an	|- ( ( A e. _om /\ B e. On ) -> ( A C_ B <-> -. B C. A ) )
14	5 13	sylibrd	|- ( ( A e. _om /\ B e. On ) -> ( A ~<_ B -> A C_ B ) )
15		ssdomg	|- ( B e. On -> ( A C_ B -> A ~<_ B ) )
16	15	adantl	|- ( ( A e. _om /\ B e. On ) -> ( A C_ B -> A ~<_ B ) )
17	14 16	impbid	|- ( ( A e. _om /\ B e. On ) -> ( A ~<_ B <-> A C_ B ) )