Theorem vpwex 5346

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 5347 from vpwex 5346. (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 4562	. 2 ⊢ 𝒫 𝑥 = {𝑤 ∣ 𝑤 ⊆ 𝑥}
2		axpow2 5336	. . . . 5 ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)
3	2	sepexi 5262	. . . 4 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥)
4		sseq1 3938	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑤 ⊆ 𝑥 ↔ 𝑧 ⊆ 𝑥))
5	4	eqabbw 2837	. . . . 5 ⊢ (𝑦 = {𝑤 ∣ 𝑤 ⊆ 𝑥} ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥))
6	5	exbii 1881	. . . 4 ⊢ (∃𝑦 𝑦 = {𝑤 ∣ 𝑤 ⊆ 𝑥} ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥))
7	3, 6	mpbir 234	. . 3 ⊢ ∃𝑦 𝑦 = {𝑤 ∣ 𝑤 ⊆ 𝑥}
8	7	issetri 3473	. 2 ⊢ {𝑤 ∣ 𝑤 ⊆ 𝑥} ∈ V
9	1, 8	eqeltri 2860	1 ⊢ 𝒫 𝑥 ∈ V

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  Vcvv 3454   ⊆ wss 3900  𝒫 cpw 4560

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pow 5334

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-pw 4562

This theorem is used by:  pwexg  5347  pwnex  7805  inf3lem7  9672  dfac8  10253  dfac13  10260  ackbij1lem8  10343  dominf  10562  numthcor  10611  dominfac  10697  intwun  10859  wunex2  10862  eltsk2g  10875  inttsk  10898  tskcard  10905  intgru  10938  gruina  10942  axgroth6  10952  ismre  17810  fnmre  17811  mreacs  17882  isacs5lem  18769  pmtrfval  19715  istopon  23285  dmtopon  23296  tgdom  23351  isfbas  24203  bj-snglex  38101  exrecfnpw  38572  pwinfi  44826  ntrrn  45383  ntrf  45384  dssmapntrcls  45389  vsetrec  51202  pgindnf  51215