Theorem nnsdomo

Description: Cardinal ordering agrees with natural number ordering. (Contributed by NM, 17-Jun-1998)

	Ref	Expression
Assertion	nnsdomo	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ≺ 𝐵 ↔ 𝐴 ⊊ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		nndomo	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵 ) )
2		nneneq	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵 ) )
3	2	notbid	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ¬ 𝐴 ≈ 𝐵 ↔ ¬ 𝐴 = 𝐵 ) )
4	1 3	anbi12d	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵 ) ↔ ( 𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵 ) ) )
5		brsdom	⊢ ( 𝐴 ≺ 𝐵 ↔ ( 𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵 ) )
6		dfpss2	⊢ ( 𝐴 ⊊ 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵 ) )
7	4 5 6	3bitr4g	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ≺ 𝐵 ↔ 𝐴 ⊊ 𝐵 ) )