Theorem setswith 4321

Description: Two ways to express the class of all sets that contain A. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
setswith	|- {x | A e. x} = if(A e. _V, ( Sk "k{{A}}), (/))

Distinct variable group:   x,A

Proof of Theorem setswith

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 4111	. . . . . . 7 |- {A} e. _V
2		opkeq1 4059	. . . . . . . 8 |- (y = {A} -> <<y, x>> = <<{A}, x>>)
3	2	eleq1d 2419	. . . . . . 7 |- (y = {A} -> (<<y, x>> e. Sk <-> <<{A}, x>> e. Sk ))
4	1, 3	rexsn 3768	. . . . . 6 |- (E.y e. {{A}}<<y, x>> e. Sk <-> <<{A}, x>> e. Sk )
5		vex 2862	. . . . . . 7 |- x e. _V
6		elssetkg 4269	. . . . . . 7 |- ((A e. _V /\ x e. _V) -> (<<{A}, x>> e. Sk <-> A e. x))
7	5, 6	mpan2 652	. . . . . 6 |- (A e. _V -> (<<{A}, x>> e. Sk <-> A e. x))
8	4, 7	syl5rbb 249	. . . . 5 |- (A e. _V -> (A e. x <-> E.y e. {{A}}<<y, x>> e. Sk ))
9	8	abbidv 2467	. . . 4 |- (A e. _V -> {x | A e. x} = {x | E.y e. {{A}}<<y, x>> e. Sk })
10		df-imak 4189	. . . 4 |- ( Sk "k{{A}}) = {x | E.y e. {{A}}<<y, x>> e. Sk }
11	9, 10	syl6eqr 2403	. . 3 |- (A e. _V -> {x | A e. x} = ( Sk "k{{A}}))
12		iftrue 3668	. . 3 |- (A e. _V -> if(A e. _V, ( Sk "k{{A}}), (/)) = ( Sk "k{{A}}))
13	11, 12	eqtr4d 2388	. 2 |- (A e. _V -> {x | A e. x} = if(A e. _V, ( Sk "k{{A}}), (/)))
14		elex 2867	. . . . . 6 |- (A e. x -> A e. _V)
15	14	con3i 127	. . . . 5 |- (-. A e. _V -> -. A e. x)
16	15	alrimiv 1631	. . . 4 |- (-. A e. _V -> A.x -. A e. x)
17		ab0 3569	. . . 4 |- ({x | A e. x} = (/) <-> A.x -. A e. x)
18	16, 17	sylibr 203	. . 3 |- (-. A e. _V -> {x | A e. x} = (/))
19		iffalse 3669	. . 3 |- (-. A e. _V -> if(A e. _V, ( Sk "k{{A}}), (/)) = (/))
20	18, 19	eqtr4d 2388	. 2 |- (-. A e. _V -> {x | A e. x} = if(A e. _V, ( Sk "k{{A}}), (/)))
21	13, 20	pm2.61i 156	1 |- {x | A e. x} = if(A e. _V, ( Sk "k{{A}}), (/))

Syntax hints:  -. wn 3   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2615  _Vcvv 2859  (/)c0 3550  ifcif 3662  {csn 3737  <<copk 4057  "_kcimak 4179   S_k cssetk 4183

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 4078  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-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-imak 4189  df-ssetk 4193

This theorem is referenced by:  setswithex  4322