Theorem ddifstab 3361

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 2232	. 2 ⊢ ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥(𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴))
2		eldif 3229	. . . . . . 7 ⊢ (𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
3		vex 2824	. . . . . . . 8 ⊢ 𝑥 ∈ V
4	3	biantrur 303	. . . . . . 7 ⊢ (¬ 𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
5		eldif 3229	. . . . . . . . 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 678	. . . . . . 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 638	. . . . . . 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 843	. . . 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 1523	. 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 842  ∀wal 1400   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∖ cdif 3217

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-stab 843  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222

This theorem is referenced by: (None)