Theorem axpweq 4768

Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4769 is not used by the proof. (Contributed by NM, 22-Jun-2009.)

Hypothesis

Ref	Expression
axpweq.1	⊢ 𝐴 ∈ V

Assertion

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

Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem axpweq

Step	Hyp	Ref	Expression
1		pwidg 4121	. . . 4 ⊢ (𝒫 𝐴 ∈ V → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴)
2		pweq 4111	. . . . . 6 ⊢ (𝑥 = 𝒫 𝐴 → 𝒫 𝑥 = 𝒫 𝒫 𝐴)
3	2	eleq2d 2673	. . . . 5 ⊢ (𝑥 = 𝒫 𝐴 → (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴))
4	3	spcegv 3267	. . . 4 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ 𝒫 𝒫 𝐴 → ∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥))
5	1, 4	mpd 15	. . 3 ⊢ (𝒫 𝐴 ∈ V → ∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥)
6		elex 3185	. . . 4 ⊢ (𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V)
7	6	exlimiv 1845	. . 3 ⊢ (∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V)
8	5, 7	impbii 198	. 2 ⊢ (𝒫 𝐴 ∈ V ↔ ∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥)
9		vex 3176	. . . . 5 ⊢ 𝑥 ∈ V
10	9	elpw2 4755	. . . 4 ⊢ (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴 ⊆ 𝑥)
11		pwss 4123	. . . . 5 ⊢ (𝒫 𝐴 ⊆ 𝑥 ↔ ∀𝑦(𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝑥))
12		dfss2 3557	. . . . . . 7 ⊢ (𝑦 ⊆ 𝐴 ↔ ∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴))
13	12	imbi1i 338	. . . . . 6 ⊢ ((𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝑥) ↔ (∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
14	13	albii 1737	. . . . 5 ⊢ (∀𝑦(𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝑥) ↔ ∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
15	11, 14	bitri 263	. . . 4 ⊢ (𝒫 𝐴 ⊆ 𝑥 ↔ ∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
16	10, 15	bitri 263	. . 3 ⊢ (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
17	16	exbii 1764	. 2 ⊢ (∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
18	8, 17	bitri 263	1 ⊢ (𝒫 𝐴 ∈ V ↔ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110

This theorem is referenced by: (None)