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Theorem ddifstab 3118
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 2079 . 2 ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥(𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴))
2 eldif 2995 . . . . . . 7 (𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
3 vex 2617 . . . . . . . 8 𝑥 ∈ V
43biantrur 297 . . . . . . 7 𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
5 eldif 2995 . . . . . . . . 9 (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
63biantrur 297 . . . . . . . . 9 𝑥𝐴 ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
75, 6bitr4i 185 . . . . . . . 8 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
87notbii 627 . . . . . . 7 𝑥 ∈ (V ∖ 𝐴) ↔ ¬ ¬ 𝑥𝐴)
92, 4, 83bitr2i 206 . . . . . 6 (𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ ¬ ¬ 𝑥𝐴)
109bibi1i 226 . . . . 5 ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴) ↔ (¬ ¬ 𝑥𝐴𝑥𝐴))
11 bi1 116 . . . . . 6 ((¬ ¬ 𝑥𝐴𝑥𝐴) → (¬ ¬ 𝑥𝐴𝑥𝐴))
12 id 19 . . . . . . 7 ((¬ ¬ 𝑥𝐴𝑥𝐴) → (¬ ¬ 𝑥𝐴𝑥𝐴))
13 notnot 592 . . . . . . 7 (𝑥𝐴 → ¬ ¬ 𝑥𝐴)
1412, 13impbid1 140 . . . . . 6 ((¬ ¬ 𝑥𝐴𝑥𝐴) → (¬ ¬ 𝑥𝐴𝑥𝐴))
1511, 14impbii 124 . . . . 5 ((¬ ¬ 𝑥𝐴𝑥𝐴) ↔ (¬ ¬ 𝑥𝐴𝑥𝐴))
1610, 15bitri 182 . . . 4 ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴) ↔ (¬ ¬ 𝑥𝐴𝑥𝐴))
17 df-stab 774 . . . 4 (STAB 𝑥𝐴 ↔ (¬ ¬ 𝑥𝐴𝑥𝐴))
1816, 17bitr4i 185 . . 3 ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴) ↔ STAB 𝑥𝐴)
1918albii 1402 . 2 (∀𝑥(𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴) ↔ ∀𝑥STAB 𝑥𝐴)
201, 19bitri 182 1 ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥STAB 𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  STAB wstab 773  wal 1285   = wceq 1287  wcel 1436  Vcvv 2614  cdif 2983
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-stab 774  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-dif 2988
This theorem is referenced by: (None)
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