Theorem axpweq 5350

Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 5364 is not used by the proof. When ax-pow 5364 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 7786). (Contributed by NM, 22-Jun-2009.)

Assertion

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

Distinct variable groups:   𝑥,𝐴,𝑦   𝑧,𝐴,𝑦

Proof of Theorem axpweq

Step	Hyp	Ref	Expression
1		pwidg 4619	. . . 4 ⊢ (𝒫 𝐴 ∈ V → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴)
2		pweq 4613	. . . . . 6 ⊢ (𝑥 = 𝒫 𝐴 → 𝒫 𝑥 = 𝒫 𝒫 𝐴)
3	2	eleq2d 2826	. . . . 5 ⊢ (𝑥 = 𝒫 𝐴 → (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴))
4	3	spcegv 3596	. . . 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 3500	. . . 4 ⊢ (𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V)
7	6	exlimiv 1929	. . 3 ⊢ (∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V)
8	5, 7	impbii 209	. 2 ⊢ (𝒫 𝐴 ∈ V ↔ ∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥)
9		vex 3483	. . . . 5 ⊢ 𝑥 ∈ V
10	9	elpw2 5333	. . . 4 ⊢ (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴 ⊆ 𝑥)
11		pwss 4622	. . . . 5 ⊢ (𝒫 𝐴 ⊆ 𝑥 ↔ ∀𝑦(𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝑥))
12		df-ss 3967	. . . . . . 7 ⊢ (𝑦 ⊆ 𝐴 ↔ ∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴))
13	12	imbi1i 349	. . . . . 6 ⊢ ((𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝑥) ↔ (∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
14	13	albii 1818	. . . . 5 ⊢ (∀𝑦(𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝑥) ↔ ∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
15	11, 14	bitri 275	. . . 4 ⊢ (𝒫 𝐴 ⊆ 𝑥 ↔ ∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
16	10, 15	bitri 275	. . 3 ⊢ (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
17	16	exbii 1847	. 2 ⊢ (∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
18	8, 17	bitri 275	1 ⊢ (𝒫 𝐴 ∈ V ↔ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1537   = wceq 1539  ∃wex 1778   ∈ wcel 2107  Vcvv 3479   ⊆ wss 3950  𝒫 cpw 4599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-in 3957  df-ss 3967  df-pw 4601

This theorem is referenced by:  grothpw  10867