Theorem axpow3

Description: A variant of the Axiom of Power Sets ax-pow . For any set x , there exists a set y whose members are exactly the subsets of x i.e. the power set of x . Axiom Pow of BellMachover p. 466. (Contributed by NM, 4-Jun-2006)

	Ref	Expression
Assertion	axpow3	⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		axpow2	⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦 )
2	1	sepexi	⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥 )
3		bicom1	⊢ ( ( 𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥 ) → ( 𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦 ) )
4	3	alimi	⊢ ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥 ) → ∀ 𝑧 ( 𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦 ) )
5	2 4	eximii	⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦 )