Theorem vpwex 5278

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 5279 from vpwex 5278. (Revised by BJ, 10-Aug-2022.)

Assertion

Ref	Expression
vpwex	⊢ 𝒫 𝑥 ∈ V

Proof of Theorem vpwex

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

Step	Hyp	Ref	Expression
1		df-pw 4541	. 2 ⊢ 𝒫 𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑥}
2		axpow2 5268	. . . . 5 ⊢ ∃𝑧∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝑧)
3	2	bm1.3ii 5206	. . . 4 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥)
4		abeq2 2945	. . . . 5 ⊢ (𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑥} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥))
5	4	exbii 1848	. . . 4 ⊢ (∃𝑧 𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑥} ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥))
6	3, 5	mpbir 233	. . 3 ⊢ ∃𝑧 𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑥}
7	6	issetri 3510	. 2 ⊢ {𝑦 ∣ 𝑦 ⊆ 𝑥} ∈ V
8	1, 7	eqeltri 2909	1 ⊢ 𝒫 𝑥 ∈ V

Syntax hints:   ↔ wb 208  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2799  Vcvv 3494   ⊆ wss 3936  𝒫 cpw 4539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-pow 5266

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-v 3496  df-in 3943  df-ss 3952  df-pw 4541

This theorem is referenced by:  pwexg  5279  pwnex  7481  inf3lem7  9097  dfac8  9561  dfac13  9568  ackbij1lem8  9649  dominf  9867  numthcor  9916  dominfac  9995  intwun  10157  wunex2  10160  eltsk2g  10173  inttsk  10196  tskcard  10203  intgru  10236  gruina  10240  axgroth6  10250  ismre  16861  fnmre  16862  mreacs  16929  isacs5lem  17779  pmtrfval  18578  istopon  21520  dmtopon  21531  tgdom  21586  isfbas  22437  bj-snglex  34288  exrecfnpw  34665  pwinfi  39943  ntrrn  40492  ntrf  40493  dssmapntrcls  40498