Theorem axpow2

Description: A variant of the Axiom of Power Sets ax-pow using subset notation. Problem in BellMachover p. 466. (Contributed by NM, 4-Jun-2006)

	Ref	Expression
Assertion	axpow2	⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		ax-pow	⊢ ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 )
2		dfss2	⊢ ( 𝑧 ⊆ 𝑥 ↔ ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥 ) )
3	2	imbi1i	⊢ ( ( 𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦 ) ↔ ( ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) )
4	3	albii	⊢ ( ∀ 𝑧 ( 𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦 ) ↔ ∀ 𝑧 ( ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) )
5	4	exbii	⊢ ( ∃ 𝑦 ∀ 𝑧 ( 𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦 ) ↔ ∃ 𝑦 ∀ 𝑧 ( ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) )
6	1 5	mpbir	⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦 )