Theorem vpwex 5348

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 5349 from vpwex 5348. (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 4564	. 2 ⊢ 𝒫 𝑥 = {𝑤 ∣ 𝑤 ⊆ 𝑥}
2		axpow2 5338	. . . . 5 ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)
3	2	sepexi 5264	. . . 4 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥)
4		sseq1 3962	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑤 ⊆ 𝑥 ↔ 𝑧 ⊆ 𝑥))
5	4	eqabbw 2836	. . . . 5 ⊢ (𝑦 = {𝑤 ∣ 𝑤 ⊆ 𝑥} ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥))
6	5	exbii 1878	. . . 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 3474	. 2 ⊢ {𝑤 ∣ 𝑤 ⊆ 𝑥} ∈ V
9	1, 8	eqeltri 2859	1 ⊢ 𝒫 𝑥 ∈ V

Syntax hints:   ↔ wb 209  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  Vcvv 3455   ⊆ wss 3905  𝒫 cpw 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pow 5336

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-ss 3922  df-pw 4564

This theorem is referenced by:  pwexg  5349  pwnex  7754  inf3lem7  9599  dfac8  10115  dfac13  10122  ackbij1lem8  10205  dominf  10424  numthcor  10473  dominfac  10553  intwun  10715  wunex2  10718  eltsk2g  10731  inttsk  10754  tskcard  10761  intgru  10794  gruina  10798  axgroth6  10808  ismre  17637  fnmre  17638  mreacs  17709  isacs5lem  18596  pmtrfval  19515  istopon  23069  dmtopon  23080  tgdom  23135  isfbas  23986  bj-snglex  37629  exrecfnpw  38047  pwinfi  44310  ntrrn  44868  ntrf  44869  dssmapntrcls  44874  vsetrec  50501  pgindnf  50514