Theorem setswithex 4323

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 4322	. 2 ⊢ {x ∣ A ∈ x} = if(A ∈ V, ( Sk “k {{A}}), ∅)
2		ssetkex 4295	. . . 4 ⊢ Sk ∈ V
3		snex 4112	. . . 4 ⊢ {{A}} ∈ V
4	2, 3	imakex 4301	. . 3 ⊢ ( Sk “k {{A}}) ∈ V
5		0ex 4111	. . 3 ⊢ ∅ ∈ V
6	4, 5	ifex 3721	. 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 2860  ∅c0 3551   ifcif 3663  {csn 3738   “_k cimak 4180   S_k cssetk 4184

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-1c 4082  ax-sset 4083  ax-si 4084  ax-typlower 4087  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-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-if 3664  df-sn 3742  df-pr 3743  df-opk 4059  df-1c 4137  df-xpk 4186  df-cnvk 4187  df-imak 4190  df-p6 4192  df-sik 4193  df-ssetk 4194

This theorem is referenced by:  nncex  4397  nnadjoinlem1  4520  spfinex  4538