New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  setswithex GIF version

Theorem setswithex 4322
 Description: The class of all sets that contain A exist. (Contributed by SF, 14-Jan-2015.)
Assertion
Ref Expression
setswithex {x A x} V
Distinct variable group:   x,A

Proof of Theorem setswithex
StepHypRef Expression
1 setswith 4321 . 2 {x A x} = if(A V, ( Skk {{A}}), )
2 ssetkex 4294 . . . 4 Sk V
3 snex 4111 . . . 4 {{A}} V
42, 3imakex 4300 . . 3 ( Skk {{A}}) V
5 0ex 4110 . . 3 V
64, 5ifex 3720 . 2 if(A V, ( Skk {{A}}), ) V
71, 6eqeltri 2423 1 {x A x} V
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1710  {cab 2339  Vcvv 2859  ∅c0 3550   ifcif 3662  {csn 3737   “k cimak 4179   Sk cssetk 4183 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-if 3663  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-xpk 4185  df-cnvk 4186  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193 This theorem is referenced by:  nncex  4396  nnadjoinlem1  4519  spfinex  4537
 Copyright terms: Public domain W3C validator