Theorem vpwex 5280

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 5281 from vpwex 5280. (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 4543	. 2 ⊢ 𝒫 𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑥}
2		axpow2 5270	. . . . 5 ⊢ ∃𝑧∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝑧)
3	2	bm1.3ii 5208	. . . 4 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥)
4		abeq2 2947	. . . . 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 3512	. 2 ⊢ {𝑦 ∣ 𝑦 ⊆ 𝑥} ∈ V
8	1, 7	eqeltri 2911	1 ⊢ 𝒫 𝑥 ∈ V

Syntax hints:   ↔ wb 208  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2801  Vcvv 3496   ⊆ wss 3938  𝒫 cpw 4541

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 2795  ax-sep 5205  ax-pow 5268

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 2802  df-cleq 2816  df-clel 2895  df-v 3498  df-in 3945  df-ss 3954  df-pw 4543

This theorem is referenced by:  pwexg  5281  pwnex  7483  inf3lem7  9099  dfac8  9563  dfac13  9570  ackbij1lem8  9651  dominf  9869  numthcor  9918  dominfac  9997  intwun  10159  wunex2  10162  eltsk2g  10175  inttsk  10198  tskcard  10205  intgru  10238  gruina  10242  axgroth6  10252  ismre  16863  fnmre  16864  mreacs  16931  isacs5lem  17781  pmtrfval  18580  istopon  21522  dmtopon  21533  tgdom  21588  isfbas  22439  bj-snglex  34287  exrecfnpw  34664  pwinfi  39930  ntrrn  40479  ntrf  40480  dssmapntrcls  40485