Theorem elom

Description: Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 . (Contributed by NM, 15-May-1994)

	Ref	Expression
Assertion	elom	⊢ ( 𝐴 ∈ ω ↔ ( 𝐴 ∈ On ∧ ∀ 𝑥 ( Lim 𝑥 → 𝐴 ∈ 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elequ1	⊢ ( 𝑦 = 𝑤 → ( 𝑦 ∈ 𝑥 ↔ 𝑤 ∈ 𝑥 ) )
2	1	imbi2d	⊢ ( 𝑦 = 𝑤 → ( ( Lim 𝑥 → 𝑦 ∈ 𝑥 ) ↔ ( Lim 𝑥 → 𝑤 ∈ 𝑥 ) ) )
3	2	albidv	⊢ ( 𝑦 = 𝑤 → ( ∀ 𝑥 ( Lim 𝑥 → 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑥 ( Lim 𝑥 → 𝑤 ∈ 𝑥 ) ) )
4		eleq1	⊢ ( 𝑤 = 𝐴 → ( 𝑤 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥 ) )
5	4	imbi2d	⊢ ( 𝑤 = 𝐴 → ( ( Lim 𝑥 → 𝑤 ∈ 𝑥 ) ↔ ( Lim 𝑥 → 𝐴 ∈ 𝑥 ) ) )
6	5	albidv	⊢ ( 𝑤 = 𝐴 → ( ∀ 𝑥 ( Lim 𝑥 → 𝑤 ∈ 𝑥 ) ↔ ∀ 𝑥 ( Lim 𝑥 → 𝐴 ∈ 𝑥 ) ) )
7		df-om	⊢ ω = { 𝑦 ∈ On ∣ ∀ 𝑥 ( Lim 𝑥 → 𝑦 ∈ 𝑥 ) }
8	3 6 7	elrab2w	⊢ ( 𝐴 ∈ ω ↔ ( 𝐴 ∈ On ∧ ∀ 𝑥 ( Lim 𝑥 → 𝐴 ∈ 𝑥 ) ) )