Theorem vpwex 4311

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 4312 from vpwex 4311. (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 3687	. 2 ⊢ 𝒫 𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑥}
2		axpow2 4308	. . . . 5 ⊢ ∃𝑧∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝑧)
3	2	bm1.3ii 4249	. . . 4 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥)
4		abeq2 2347	. . . . 5 ⊢ (𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑥} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥))
5	4	exbii 1658	. . . 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 2831	. 2 ⊢ {𝑦 ∣ 𝑦 ⊆ 𝑥} ∈ V
8	1, 7	eqeltri 2311	1 ⊢ 𝒫 𝑥 ∈ V

Syntax hints:   ↔ wb 105  ∀wal 1400   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {cab 2224  Vcvv 2821   ⊆ wss 3220  𝒫 cpw 3685

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687

This theorem is referenced by:  pwexg  4312  pwnex  4590  exmidpw2en  7209  metuex  14864  istopon  15037  dmtopon  15047  tgdom  15096