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Theorem ddifstab 3267
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 𝑥𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem ddifstab
StepHypRef Expression
1 dfcleq 2171 . 2 ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥(𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴))
2 eldif 3138 . . . . . . 7 (𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
3 vex 2740 . . . . . . . 8 𝑥 ∈ V
43biantrur 303 . . . . . . 7 𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
5 eldif 3138 . . . . . . . . 9 (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
63biantrur 303 . . . . . . . . 9 𝑥𝐴 ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
75, 6bitr4i 187 . . . . . . . 8 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
87notbii 668 . . . . . . 7 𝑥 ∈ (V ∖ 𝐴) ↔ ¬ ¬ 𝑥𝐴)
92, 4, 83bitr2i 208 . . . . . 6 (𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ ¬ ¬ 𝑥𝐴)
109bibi1i 228 . . . . 5 ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴) ↔ (¬ ¬ 𝑥𝐴𝑥𝐴))
11 biimp 118 . . . . . 6 ((¬ ¬ 𝑥𝐴𝑥𝐴) → (¬ ¬ 𝑥𝐴𝑥𝐴))
12 id 19 . . . . . . 7 ((¬ ¬ 𝑥𝐴𝑥𝐴) → (¬ ¬ 𝑥𝐴𝑥𝐴))
13 notnot 629 . . . . . . 7 (𝑥𝐴 → ¬ ¬ 𝑥𝐴)
1412, 13impbid1 142 . . . . . 6 ((¬ ¬ 𝑥𝐴𝑥𝐴) → (¬ ¬ 𝑥𝐴𝑥𝐴))
1511, 14impbii 126 . . . . 5 ((¬ ¬ 𝑥𝐴𝑥𝐴) ↔ (¬ ¬ 𝑥𝐴𝑥𝐴))
1610, 15bitri 184 . . . 4 ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴) ↔ (¬ ¬ 𝑥𝐴𝑥𝐴))
17 df-stab 831 . . . 4 (STAB 𝑥𝐴 ↔ (¬ ¬ 𝑥𝐴𝑥𝐴))
1816, 17bitr4i 187 . . 3 ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴) ↔ STAB 𝑥𝐴)
1918albii 1470 . 2 (∀𝑥(𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴) ↔ ∀𝑥STAB 𝑥𝐴)
201, 19bitri 184 1 ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥STAB 𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  STAB wstab 830  wal 1351   = wceq 1353  wcel 2148  Vcvv 2737  cdif 3126
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-stab 831  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131
This theorem is referenced by: (None)
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