Theorem cnvkex 4288

Description: The Kuratowski converse of a set is a set. (Contributed by SF, 14-Jan-2015.)

Hypothesis

Ref	Expression
cnvkex.1	⊢ A ∈ V

Assertion

Ref	Expression
cnvkex	⊢ ◡kA ∈ V

Proof of Theorem cnvkex

Step	Hyp	Ref	Expression
1		cnvkex.1	. 2 ⊢ A ∈ V
2		cnvkexg 4287	. 2 ⊢ (A ∈ V → ◡kA ∈ V)
3	1, 2	ax-mp 5	1 ⊢ ◡kA ∈ V

Syntax hints:   ∈ wcel 1710  Vcvv 2860  ^◡_kccnvk 4176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059  df-xpk 4186  df-cnvk 4187

This theorem is referenced by:  idkex  4315  uniexg  4317  intexg  4320  nncaddccl  4420  nnsucelrlem1  4425  preaddccan2lem1  4455  ltfintrilem1  4466  ncfinlowerlem1  4483  tfinrelkex  4488  oddfinex  4505  evenodddisjlem1  4516  nnpweqlem1  4523  sfintfinlem1  4532  tfinnnlem1  4534  vfinspclt  4553  opexg  4588  proj1exg  4592  proj2exg  4593  phialllem1  4617  setconslem5  4736  1stex  4740  swapex  4743  ssetex  4745  coexg  4750  siexg  4753