Theorem nndomo

Description: Cardinal ordering agrees with natural number ordering. Example 3 of Enderton p. 146. (Contributed by NM, 17-Jun-1998)

	Ref	Expression
Assertion	nndomo	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		nnon	⊢ ( 𝐵 ∈ ω → 𝐵 ∈ On )
2		nndomog	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ On ) → ( 𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵 ) )
3	1 2	sylan2	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵 ) )