Theorem vpwex 5327

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 5328 from vpwex 5327. (Revised by BJ, 10-Aug-2022.)

Assertion

Ref	Expression
vpwex	⊢ 𝒫 𝑥 ∈ V

Proof of Theorem vpwex

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

Step	Hyp	Ref	Expression
1		df-pw 4561	. 2 ⊢ 𝒫 𝑥 = {𝑤 ∣ 𝑤 ⊆ 𝑥}
2		axpow2 5317	. . . . 5 ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)
3	2	sepexi 5251	. . . 4 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥)
4		sseq1 3969	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑤 ⊆ 𝑥 ↔ 𝑧 ⊆ 𝑥))
5	4	eqabbw 2802	. . . . 5 ⊢ (𝑦 = {𝑤 ∣ 𝑤 ⊆ 𝑥} ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥))
6	5	exbii 1848	. . . 4 ⊢ (∃𝑦 𝑦 = {𝑤 ∣ 𝑤 ⊆ 𝑥} ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥))
7	3, 6	mpbir 231	. . 3 ⊢ ∃𝑦 𝑦 = {𝑤 ∣ 𝑤 ⊆ 𝑥}
8	7	issetri 3463	. 2 ⊢ {𝑤 ∣ 𝑤 ⊆ 𝑥} ∈ V
9	1, 8	eqeltri 2824	1 ⊢ 𝒫 𝑥 ∈ V

Syntax hints:   ↔ wb 206  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  Vcvv 3444   ⊆ wss 3911  𝒫 cpw 4559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-pow 5315

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3446  df-ss 3928  df-pw 4561

This theorem is referenced by:  pwexg  5328  pwnex  7715  inf3lem7  9563  dfac8  10065  dfac13  10072  ackbij1lem8  10155  dominf  10374  numthcor  10423  dominfac  10502  intwun  10664  wunex2  10667  eltsk2g  10680  inttsk  10703  tskcard  10710  intgru  10743  gruina  10747  axgroth6  10757  ismre  17527  fnmre  17528  mreacs  17599  isacs5lem  18486  pmtrfval  19364  istopon  22832  dmtopon  22843  tgdom  22898  isfbas  23749  bj-snglex  36954  exrecfnpw  37362  pwinfi  43546  ntrrn  44104  ntrf  44105  dssmapntrcls  44110  vsetrec  49685  pgindnf  49698