Theorem ddifstab 3291

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 2187	. 2 ⊢ ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥(𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴))
2		eldif 3162	. . . . . . 7 ⊢ (𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
3		vex 2763	. . . . . . . 8 ⊢ 𝑥 ∈ V
4	3	biantrur 303	. . . . . . 7 ⊢ (¬ 𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
5		eldif 3162	. . . . . . . . 9 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐴))
6	3	biantrur 303	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐴))
7	5, 6	bitr4i 187	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴)
8	7	notbii 669	. . . . . . 7 ⊢ (¬ 𝑥 ∈ (V ∖ 𝐴) ↔ ¬ ¬ 𝑥 ∈ 𝐴)
9	2, 4, 8	3bitr2i 208	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ ¬ ¬ 𝑥 ∈ 𝐴)
10	9	bibi1i 228	. . . . 5 ⊢ ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴) ↔ (¬ ¬ 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
11		biimp 118	. . . . . 6 ⊢ ((¬ ¬ 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) → (¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
12		id 19	. . . . . . 7 ⊢ ((¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴) → (¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
13		notnot 630	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ¬ ¬ 𝑥 ∈ 𝐴)
14	12, 13	impbid1 142	. . . . . 6 ⊢ ((¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴) → (¬ ¬ 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
15	11, 14	impbii 126	. . . . 5 ⊢ ((¬ ¬ 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) ↔ (¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
16	10, 15	bitri 184	. . . 4 ⊢ ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴) ↔ (¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
17		df-stab 832	. . . 4 ⊢ (STAB 𝑥 ∈ 𝐴 ↔ (¬ ¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
18	16, 17	bitr4i 187	. . 3 ⊢ ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴) ↔ STAB 𝑥 ∈ 𝐴)
19	18	albii 1481	. 2 ⊢ (∀𝑥(𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴) ↔ ∀𝑥STAB 𝑥 ∈ 𝐴)
20	1, 19	bitri 184	1 ⊢ ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥STAB 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  STAB wstab 831  ∀wal 1362   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ∖ cdif 3150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-stab 832  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155

This theorem is referenced by: (None)