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 ∈ ω ∧ 𝐵 ∈ On ) → ( 𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		php2	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴 ) → 𝐵 ≺ 𝐴 )
2	1	ex	⊢ ( 𝐴 ∈ ω → ( 𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝐴 ) )
3		domnsym	⊢ ( 𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴 )
4	2 3	nsyli	⊢ ( 𝐴 ∈ ω → ( 𝐴 ≼ 𝐵 → ¬ 𝐵 ⊊ 𝐴 ) )
5	4	adantr	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ On ) → ( 𝐴 ≼ 𝐵 → ¬ 𝐵 ⊊ 𝐴 ) )
6		nnord	⊢ ( 𝐴 ∈ ω → Ord 𝐴 )
7		eloni	⊢ ( 𝐵 ∈ On → Ord 𝐵 )
8		ordtri1	⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴 ) )
9		ordelpss	⊢ ( ( Ord 𝐵 ∧ Ord 𝐴 ) → ( 𝐵 ∈ 𝐴 ↔ 𝐵 ⊊ 𝐴 ) )
10	9	ancoms	⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝐵 ∈ 𝐴 ↔ 𝐵 ⊊ 𝐴 ) )
11	10	notbid	⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( ¬ 𝐵 ∈ 𝐴 ↔ ¬ 𝐵 ⊊ 𝐴 ) )
12	8 11	bitrd	⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ⊊ 𝐴 ) )
13	6 7 12	syl2an	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ⊊ 𝐴 ) )
14	5 13	sylibrd	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ On ) → ( 𝐴 ≼ 𝐵 → 𝐴 ⊆ 𝐵 ) )
15		ssdomg	⊢ ( 𝐵 ∈ On → ( 𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵 ) )
16	15	adantl	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵 ) )
17	14 16	impbid	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ On ) → ( 𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵 ) )