Theorem vpwex 5243

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 5244 from vpwex 5243. (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 4499	. 2 ⊢ 𝒫 𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑥}
2		axpow2 5233	. . . . 5 ⊢ ∃𝑧∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝑧)
3	2	bm1.3ii 5170	. . . 4 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥)
4		abeq2 2922	. . . . 5 ⊢ (𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑥} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥))
5	4	exbii 1849	. . . 4 ⊢ (∃𝑧 𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑥} ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥))
6	3, 5	mpbir 234	. . 3 ⊢ ∃𝑧 𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑥}
7	6	issetri 3457	. 2 ⊢ {𝑦 ∣ 𝑦 ⊆ 𝑥} ∈ V
8	1, 7	eqeltri 2886	1 ⊢ 𝒫 𝑥 ∈ V

Syntax hints:   ↔ wb 209  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  Vcvv 3441   ⊆ wss 3881  𝒫 cpw 4497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-pow 5231

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-pw 4499

This theorem is referenced by:  pwexg  5244  pwnex  7461  inf3lem7  9081  dfac8  9546  dfac13  9553  ackbij1lem8  9638  dominf  9856  numthcor  9905  dominfac  9984  intwun  10146  wunex2  10149  eltsk2g  10162  inttsk  10185  tskcard  10192  intgru  10225  gruina  10229  axgroth6  10239  ismre  16853  fnmre  16854  mreacs  16921  isacs5lem  17771  pmtrfval  18570  istopon  21517  dmtopon  21528  tgdom  21583  isfbas  22434  bj-snglex  34409  exrecfnpw  34798  pwinfi  40263  ntrrn  40825  ntrf  40826  dssmapntrcls  40831