Theorem ddifstab 3254

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 2159	. 2 ⊢ ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥(𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴))
2		eldif 3125	. . . . . . 7 ⊢ (𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
3		vex 2729	. . . . . . . 8 ⊢ 𝑥 ∈ V
4	3	biantrur 301	. . . . . . 7 ⊢ (¬ 𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
5		eldif 3125	. . . . . . . . 9 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐴))
6	3	biantrur 301	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐴))
7	5, 6	bitr4i 186	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴)
8	7	notbii 658	. . . . . . 7 ⊢ (¬ 𝑥 ∈ (V ∖ 𝐴) ↔ ¬ ¬ 𝑥 ∈ 𝐴)
9	2, 4, 8	3bitr2i 207	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ ¬ ¬ 𝑥 ∈ 𝐴)
10	9	bibi1i 227	. . . . 5 ⊢ ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴) ↔ (¬ ¬ 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
11		biimp 117	. . . . . 6 ⊢ ((¬ ¬ 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) → (¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
12		id 19	. . . . . . 7 ⊢ ((¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴) → (¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
13		notnot 619	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ¬ ¬ 𝑥 ∈ 𝐴)
14	12, 13	impbid1 141	. . . . . 6 ⊢ ((¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴) → (¬ ¬ 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
15	11, 14	impbii 125	. . . . 5 ⊢ ((¬ ¬ 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) ↔ (¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
16	10, 15	bitri 183	. . . 4 ⊢ ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴) ↔ (¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
17		df-stab 821	. . . 4 ⊢ (STAB 𝑥 ∈ 𝐴 ↔ (¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
18	16, 17	bitr4i 186	. . 3 ⊢ ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴) ↔ STAB 𝑥 ∈ 𝐴)
19	18	albii 1458	. 2 ⊢ (∀𝑥(𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴) ↔ ∀𝑥STAB 𝑥 ∈ 𝐴)
20	1, 19	bitri 183	1 ⊢ ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥STAB 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  STAB wstab 820  ∀wal 1341   = wceq 1343   ∈ wcel 2136  Vcvv 2726   ∖ cdif 3113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-stab 821  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118

This theorem is referenced by: (None)