Theorem axpweq 4150

Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4153 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 3573	. . . 4 ⊢ (𝒫 𝐴 ∈ V → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴)
2		pweq 3562	. . . . . 6 ⊢ (𝑥 = 𝒫 𝐴 → 𝒫 𝑥 = 𝒫 𝒫 𝐴)
3	2	eleq2d 2236	. . . . 5 ⊢ (𝑥 = 𝒫 𝐴 → (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴))
4	3	spcegv 2814	. . . 4 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ 𝒫 𝒫 𝐴 → ∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥))
5	1, 4	mpd 13	. . 3 ⊢ (𝒫 𝐴 ∈ V → ∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥)
6		elex 2737	. . . 4 ⊢ (𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V)
7	6	exlimiv 1586	. . 3 ⊢ (∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V)
8	5, 7	impbii 125	. 2 ⊢ (𝒫 𝐴 ∈ V ↔ ∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥)
9		vex 2729	. . . . 5 ⊢ 𝑥 ∈ V
10	9	elpw2 4136	. . . 4 ⊢ (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴 ⊆ 𝑥)
11		pwss 3575	. . . . 5 ⊢ (𝒫 𝐴 ⊆ 𝑥 ↔ ∀𝑦(𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝑥))
12		dfss2 3131	. . . . . . 7 ⊢ (𝑦 ⊆ 𝐴 ↔ ∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴))
13	12	imbi1i 237	. . . . . 6 ⊢ ((𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝑥) ↔ (∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
14	13	albii 1458	. . . . 5 ⊢ (∀𝑦(𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝑥) ↔ ∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
15	11, 14	bitri 183	. . . 4 ⊢ (𝒫 𝐴 ⊆ 𝑥 ↔ ∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
16	10, 15	bitri 183	. . 3 ⊢ (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
17	16	exbii 1593	. 2 ⊢ (∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
18	8, 17	bitri 183	1 ⊢ (𝒫 𝐴 ∈ V ↔ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Vcvv 2726   ⊆ wss 3116  𝒫 cpw 3559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-sep 4100

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561

This theorem is referenced by: (None)