Theorem infn0

Description: An infinite set is not empty. (Contributed by NM, 23-Oct-2004)

	Ref	Expression
Assertion	infn0	⊢ ( ω ≼ 𝐴 → 𝐴 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		peano1	⊢ ∅ ∈ ω
2		infsdomnn	⊢ ( ( ω ≼ 𝐴 ∧ ∅ ∈ ω ) → ∅ ≺ 𝐴 )
3	1 2	mpan2	⊢ ( ω ≼ 𝐴 → ∅ ≺ 𝐴 )
4		reldom	⊢ Rel ≼
5	4	brrelex2i	⊢ ( ω ≼ 𝐴 → 𝐴 ∈ V )
6		0sdomg	⊢ ( 𝐴 ∈ V → ( ∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅ ) )
7	5 6	syl	⊢ ( ω ≼ 𝐴 → ( ∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅ ) )
8	3 7	mpbid	⊢ ( ω ≼ 𝐴 → 𝐴 ≠ ∅ )