Theorem ackbij1lem3

Description: Lemma for ackbij2 . (Contributed by Stefan O'Rear, 18-Nov-2014)

	Ref	Expression
Assertion	ackbij1lem3	⊢ ( 𝐴 ∈ ω → 𝐴 ∈ ( 𝒫 ω ∩ Fin ) )

Proof

Step	Hyp	Ref	Expression
1		ordom	⊢ Ord ω
2		ordelss	⊢ ( ( Ord ω ∧ 𝐴 ∈ ω ) → 𝐴 ⊆ ω )
3	1 2	mpan	⊢ ( 𝐴 ∈ ω → 𝐴 ⊆ ω )
4		elpwg	⊢ ( 𝐴 ∈ ω → ( 𝐴 ∈ 𝒫 ω ↔ 𝐴 ⊆ ω ) )
5	3 4	mpbird	⊢ ( 𝐴 ∈ ω → 𝐴 ∈ 𝒫 ω )
6		nnfi	⊢ ( 𝐴 ∈ ω → 𝐴 ∈ Fin )
7	5 6	elind	⊢ ( 𝐴 ∈ ω → 𝐴 ∈ ( 𝒫 ω ∩ Fin ) )