Theorem pwex 3953

Description: Power set axiom expressed in class notation. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Hypothesis

Ref	Expression
zfpowcl.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
pwex	⊢ 𝒫 𝐴 ∈ V

Proof of Theorem pwex

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfpowcl.1	. 2 ⊢ 𝐴 ∈ V
2		pweq 3385	. . 3 ⊢ (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴)
3	2	eleq1d 2147	. 2 ⊢ (𝑧 = 𝐴 → (𝒫 𝑧 ∈ V ↔ 𝒫 𝐴 ∈ V))
4		df-pw 3384	. . 3 ⊢ 𝒫 𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑧}
5		axpow2 3950	. . . . . 6 ⊢ ∃𝑥∀𝑦(𝑦 ⊆ 𝑧 → 𝑦 ∈ 𝑥)
6	5	bm1.3ii 3899	. . . . 5 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ⊆ 𝑧)
7		abeq2 2187	. . . . . 6 ⊢ (𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑧} ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ⊆ 𝑧))
8	7	exbii 1536	. . . . 5 ⊢ (∃𝑥 𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑧} ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ⊆ 𝑧))
9	6, 8	mpbir 144	. . . 4 ⊢ ∃𝑥 𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑧}
10	9	issetri 2608	. . 3 ⊢ {𝑦 ∣ 𝑦 ⊆ 𝑧} ∈ V
11	4, 10	eqeltri 2151	. 2 ⊢ 𝒫 𝑧 ∈ V
12	1, 3, 11	vtocl 2653	1 ⊢ 𝒫 𝐴 ∈ V

Syntax hints:   ↔ wb 103  ∀wal 1282   = wceq 1284  ∃wex 1421   ∈ wcel 1433  {cab 2067  Vcvv 2601   ⊆ wss 2973  𝒫 cpw 3382

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384

This theorem is referenced by:  pwexg  3954  p0ex  3959  pp0ex  3960  ord3ex  3961  abexssex  5772  npex  6663  axcnex  7027  pnfxr  8846  mnfxr  8848  ixxex  8922