Theorem axpweq

Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow is not used by the proof. When ax-pow is assumed and A is a set, both sides of the biconditional hold. In ZF, both sides hold if and only if A is a set (see pwexr ). (Contributed by NM, 22-Jun-2009)

	Ref	Expression
Assertion	axpweq	⊢ ( 𝒫 𝐴 ∈ V ↔ ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴 ) → 𝑦 ∈ 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		pwidg	⊢ ( 𝒫 𝐴 ∈ V → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴 )
2		pweq	⊢ ( 𝑥 = 𝒫 𝐴 → 𝒫 𝑥 = 𝒫 𝒫 𝐴 )
3	2	eleq2d	⊢ ( 𝑥 = 𝒫 𝐴 → ( 𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴 ) )
4	3	spcegv	⊢ ( 𝒫 𝐴 ∈ V → ( 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴 → ∃ 𝑥 𝒫 𝐴 ∈ 𝒫 𝑥 ) )
5	1 4	mpd	⊢ ( 𝒫 𝐴 ∈ V → ∃ 𝑥 𝒫 𝐴 ∈ 𝒫 𝑥 )
6		elex	⊢ ( 𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V )
7	6	exlimiv	⊢ ( ∃ 𝑥 𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V )
8	5 7	impbii	⊢ ( 𝒫 𝐴 ∈ V ↔ ∃ 𝑥 𝒫 𝐴 ∈ 𝒫 𝑥 )
9		vex	⊢ 𝑥 ∈ V
10	9	elpw2	⊢ ( 𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴 ⊆ 𝑥 )
11		pwss	⊢ ( 𝒫 𝐴 ⊆ 𝑥 ↔ ∀ 𝑦 ( 𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝑥 ) )
12		dfss2	⊢ ( 𝑦 ⊆ 𝐴 ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴 ) )
13	12	imbi1i	⊢ ( ( 𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝑥 ) ↔ ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴 ) → 𝑦 ∈ 𝑥 ) )
14	13	albii	⊢ ( ∀ 𝑦 ( 𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑦 ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴 ) → 𝑦 ∈ 𝑥 ) )
15	11 14	bitri	⊢ ( 𝒫 𝐴 ⊆ 𝑥 ↔ ∀ 𝑦 ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴 ) → 𝑦 ∈ 𝑥 ) )
16	10 15	bitri	⊢ ( 𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∀ 𝑦 ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴 ) → 𝑦 ∈ 𝑥 ) )
17	16	exbii	⊢ ( ∃ 𝑥 𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴 ) → 𝑦 ∈ 𝑥 ) )
18	8 17	bitri	⊢ ( 𝒫 𝐴 ∈ V ↔ ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴 ) → 𝑦 ∈ 𝑥 ) )