Theorem ddifstab 3104

Description: A class is equal to its double complement if and only if it is stable (that is, membership in it is a stable property). (Contributed by BJ, 12-Dec-2021.)

Assertion

Ref	Expression
ddifstab	⊢ ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥STAB 𝑥 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem ddifstab

Step	Hyp	Ref	Expression
1		dfcleq 2075	. 2 ⊢ ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥(𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴))
2		eldif 2982	. . . . . . 7 ⊢ (𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
3		vex 2604	. . . . . . . 8 ⊢ 𝑥 ∈ V
4	3	biantrur 297	. . . . . . 7 ⊢ (¬ 𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
5		eldif 2982	. . . . . . . . 9 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐴))
6	3	biantrur 297	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐴))
7	5, 6	bitr4i 185	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴)
8	7	notbii 626	. . . . . . 7 ⊢ (¬ 𝑥 ∈ (V ∖ 𝐴) ↔ ¬ ¬ 𝑥 ∈ 𝐴)
9	2, 4, 8	3bitr2i 206	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ ¬ ¬ 𝑥 ∈ 𝐴)
10	9	bibi1i 226	. . . . 5 ⊢ ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴) ↔ (¬ ¬ 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
11		bi1 116	. . . . . 6 ⊢ ((¬ ¬ 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) → (¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
12		id 19	. . . . . . 7 ⊢ ((¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴) → (¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
13		notnot 591	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ¬ ¬ 𝑥 ∈ 𝐴)
14	12, 13	impbid1 140	. . . . . 6 ⊢ ((¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴) → (¬ ¬ 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
15	11, 14	impbii 124	. . . . 5 ⊢ ((¬ ¬ 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) ↔ (¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
16	10, 15	bitri 182	. . . 4 ⊢ ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴) ↔ (¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
17		df-stab 773	. . . 4 ⊢ (STAB 𝑥 ∈ 𝐴 ↔ (¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
18	16, 17	bitr4i 185	. . 3 ⊢ ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴) ↔ STAB 𝑥 ∈ 𝐴)
19	18	albii 1399	. 2 ⊢ (∀𝑥(𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴) ↔ ∀𝑥STAB 𝑥 ∈ 𝐴)
20	1, 19	bitri 182	1 ⊢ ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥STAB 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103  STAB wstab 772  ∀wal 1282   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ∖ cdif 2970

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-stab 773  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975

This theorem is referenced by: (None)