Theorem vpwex 5342

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 5343 from vpwex 5342. (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 4559	. 2 ⊢ 𝒫 𝑥 = {𝑤 ∣ 𝑤 ⊆ 𝑥}
2		axpow2 5332	. . . . 5 ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)
3	2	sepexi 5258	. . . 4 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥)
4		sseq1 3956	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑤 ⊆ 𝑥 ↔ 𝑧 ⊆ 𝑥))
5	4	eqabbw 2833	. . . . 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 3469	. 2 ⊢ {𝑤 ∣ 𝑤 ⊆ 𝑥} ∈ V
9	1, 8	eqeltri 2856	1 ⊢ 𝒫 𝑥 ∈ V

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738  Vcvv 3450   ⊆ wss 3899  𝒫 cpw 4557

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pow 5330

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3916  df-pw 4559

This theorem is used by:  pwexg  5343  pwnex  7759  inf3lem7  9616  dfac8  10141  dfac13  10148  ackbij1lem8  10231  dominf  10450  numthcor  10499  dominfac  10585  intwun  10747  wunex2  10750  eltsk2g  10763  inttsk  10786  tskcard  10793  intgru  10826  gruina  10830  axgroth6  10840  ismre  17677  fnmre  17678  mreacs  17749  isacs5lem  18636  pmtrfval  19580  istopon  23140  dmtopon  23151  tgdom  23206  isfbas  24058  bj-snglex  37720  exrecfnpw  38138  pwinfi  44407  ntrrn  44965  ntrf  44966  dssmapntrcls  44971  vsetrec  50632  pgindnf  50645