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 ∈ x} = if(A ∈ 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} ∈ V
2		opkeq1 4059	. . . . . . . 8 ⊢ (y = {A} → ⟪y, x⟫ = ⟪{A}, x⟫)
3	2	eleq1d 2419	. . . . . . 7 ⊢ (y = {A} → (⟪y, x⟫ ∈ Sk ↔ ⟪{A}, x⟫ ∈ Sk ))
4	1, 3	rexsn 3768	. . . . . 6 ⊢ (∃y ∈ {{A}}⟪y, x⟫ ∈ Sk ↔ ⟪{A}, x⟫ ∈ Sk )
5		vex 2862	. . . . . . 7 ⊢ x ∈ V
6		elssetkg 4269	. . . . . . 7 ⊢ ((A ∈ V ∧ x ∈ V) → (⟪{A}, x⟫ ∈ Sk ↔ A ∈ x))
7	5, 6	mpan2 652	. . . . . 6 ⊢ (A ∈ V → (⟪{A}, x⟫ ∈ Sk ↔ A ∈ x))
8	4, 7	syl5rbb 249	. . . . 5 ⊢ (A ∈ V → (A ∈ x ↔ ∃y ∈ {{A}}⟪y, x⟫ ∈ Sk ))
9	8	abbidv 2467	. . . 4 ⊢ (A ∈ V → {x ∣ A ∈ x} = {x ∣ ∃y ∈ {{A}}⟪y, x⟫ ∈ Sk })
10		df-imak 4189	. . . 4 ⊢ ( Sk “k {{A}}) = {x ∣ ∃y ∈ {{A}}⟪y, x⟫ ∈ Sk }
11	9, 10	syl6eqr 2403	. . 3 ⊢ (A ∈ V → {x ∣ A ∈ x} = ( Sk “k {{A}}))
12		iftrue 3668	. . 3 ⊢ (A ∈ V → if(A ∈ V, ( Sk “k {{A}}), ∅) = ( Sk “k {{A}}))
13	11, 12	eqtr4d 2388	. 2 ⊢ (A ∈ V → {x ∣ A ∈ x} = if(A ∈ V, ( Sk “k {{A}}), ∅))
14		elex 2867	. . . . . 6 ⊢ (A ∈ x → A ∈ V)
15	14	con3i 127	. . . . 5 ⊢ (¬ A ∈ V → ¬ A ∈ x)
16	15	alrimiv 1631	. . . 4 ⊢ (¬ A ∈ V → ∀x ¬ A ∈ x)
17		ab0 3569	. . . 4 ⊢ ({x ∣ A ∈ x} = ∅ ↔ ∀x ¬ A ∈ x)
18	16, 17	sylibr 203	. . 3 ⊢ (¬ A ∈ V → {x ∣ A ∈ x} = ∅)
19		iffalse 3669	. . 3 ⊢ (¬ A ∈ V → if(A ∈ V, ( Sk “k {{A}}), ∅) = ∅)
20	18, 19	eqtr4d 2388	. 2 ⊢ (¬ A ∈ V → {x ∣ A ∈ x} = if(A ∈ V, ( Sk “k {{A}}), ∅))
21	13, 20	pm2.61i 156	1 ⊢ {x ∣ A ∈ x} = if(A ∈ V, ( Sk “k {{A}}), ∅)

Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859  ∅c0 3550   ifcif 3662  {csn 3737  ⟪copk 4057   “_k cimak 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