Theorem infsdomnn

Description: An infinite set strictly dominates a natural number. (Contributed by NM, 22-Nov-2004) (Revised by Mario Carneiro, 27-Apr-2015)

	Ref	Expression
Assertion	infsdomnn	⊢ ( ( ω ≼ 𝐴 ∧ 𝐵 ∈ ω ) → 𝐵 ≺ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		reldom	⊢ Rel ≼
2	1	brrelex1i	⊢ ( ω ≼ 𝐴 → ω ∈ V )
3		nnsdomg	⊢ ( ( ω ∈ V ∧ 𝐵 ∈ ω ) → 𝐵 ≺ ω )
4	2 3	sylan	⊢ ( ( ω ≼ 𝐴 ∧ 𝐵 ∈ ω ) → 𝐵 ≺ ω )
5		simpl	⊢ ( ( ω ≼ 𝐴 ∧ 𝐵 ∈ ω ) → ω ≼ 𝐴 )
6		sdomdomtr	⊢ ( ( 𝐵 ≺ ω ∧ ω ≼ 𝐴 ) → 𝐵 ≺ 𝐴 )
7	4 5 6	syl2anc	⊢ ( ( ω ≼ 𝐴 ∧ 𝐵 ∈ ω ) → 𝐵 ≺ 𝐴 )