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

Step	Hyp	Ref	Expression
1		setswith 4321	. 2 ⊢ {x ∣ A ∈ x} = if(A ∈ V, ( Sk “k {{A}}), ∅)
2		ssetkex 4294	. . . 4 ⊢ Sk ∈ V
3		snex 4111	. . . 4 ⊢ {{A}} ∈ V
4	2, 3	imakex 4300	. . 3 ⊢ ( Sk “k {{A}}) ∈ V
5		0ex 4110	. . 3 ⊢ ∅ ∈ V
6	4, 5	ifex 3720	. 2 ⊢ if(A ∈ V, ( Sk “k {{A}}), ∅) ∈ V
7	1, 6	eqeltri 2423	1 ⊢ {x ∣ A ∈ x} ∈ V

Syntax hints:   ∈ wcel 1710  {cab 2339  Vcvv 2859  ∅c0 3550   ifcif 3662  {csn 3737   “_k cimak 4179   S_k 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