Theorem vpwex 5306

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 5307 from vpwex 5306. (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 4531	. 2 ⊢ 𝒫 𝑥 = {𝑤 ∣ 𝑤 ⊆ 𝑥}
2		axpow2 5296	. . . . 5 ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)
3	2	sepexi 5223	. . . 4 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥)
4		sseq1 3940	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑤 ⊆ 𝑥 ↔ 𝑧 ⊆ 𝑥))
5	4	eqabbw 2812	. . . . 5 ⊢ (𝑦 = {𝑤 ∣ 𝑤 ⊆ 𝑥} ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥))
6	5	exbii 1855	. . . 4 ⊢ (∃𝑦 𝑦 = {𝑤 ∣ 𝑤 ⊆ 𝑥} ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥))
7	3, 6	mpbir 232	. . 3 ⊢ ∃𝑦 𝑦 = {𝑤 ∣ 𝑤 ⊆ 𝑥}
8	7	issetri 3450	. 2 ⊢ {𝑤 ∣ 𝑤 ⊆ 𝑥} ∈ V
9	1, 8	eqeltri 2835	1 ⊢ 𝒫 𝑥 ∈ V

Syntax hints:   ↔ wb 207  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717  Vcvv 3431   ⊆ wss 3883  𝒫 cpw 4529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pow 5294

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-ss 3900  df-pw 4531

This theorem is referenced by:  pwexg  5307  pwnex  7702  inf3lem7  9546  dfac8  10049  dfac13  10056  ackbij1lem8  10139  dominf  10358  numthcor  10407  dominfac  10487  intwun  10649  wunex2  10652  eltsk2g  10665  inttsk  10688  tskcard  10695  intgru  10728  gruina  10732  axgroth6  10742  ismre  17543  fnmre  17544  mreacs  17615  isacs5lem  18502  pmtrfval  19416  istopon  22895  dmtopon  22906  tgdom  22961  isfbas  23812  bj-snglex  37326  exrecfnpw  37743  pwinfi  44008  ntrrn  44566  ntrf  44567  dssmapntrcls  44572  vsetrec  50193  pgindnf  50206