Theorem vpwex 4239

Description: Power set axiom: the powerclass of a set is a set. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Revised to prove pwexg 4240 from vpwex 4239. (Revised by BJ, 10-Aug-2022.)

Assertion

Ref	Expression
vpwex	⊢ 𝒫 𝑥 ∈ V

Proof of Theorem vpwex

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

Step	Hyp	Ref	Expression
1		df-pw 3628	. 2 ⊢ 𝒫 𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑥}
2		axpow2 4236	. . . . 5 ⊢ ∃𝑧∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝑧)
3	2	bm1.3ii 4181	. . . 4 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥)
4		abeq2 2316	. . . . 5 ⊢ (𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑥} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥))
5	4	exbii 1629	. . . 4 ⊢ (∃𝑧 𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑥} ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥))
6	3, 5	mpbir 146	. . 3 ⊢ ∃𝑧 𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑥}
7	6	issetri 2786	. 2 ⊢ {𝑦 ∣ 𝑦 ⊆ 𝑥} ∈ V
8	1, 7	eqeltri 2280	1 ⊢ 𝒫 𝑥 ∈ V

Syntax hints:   ↔ wb 105  ∀wal 1371   = wceq 1373  ∃wex 1516   ∈ wcel 2178  {cab 2193  Vcvv 2776   ⊆ wss 3174  𝒫 cpw 3626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-v 2778  df-in 3180  df-ss 3187  df-pw 3628

This theorem is referenced by:  pwexg  4240  pwnex  4514  exmidpw2en  7035  metuex  14432  istopon  14600  dmtopon  14610  tgdom  14659