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Theorem ddifstab 3239
 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 2151 . 2 ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥(𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴))
2 eldif 3111 . . . . . . 7 (𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
3 vex 2715 . . . . . . . 8 𝑥 ∈ V
43biantrur 301 . . . . . . 7 𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
5 eldif 3111 . . . . . . . . 9 (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
63biantrur 301 . . . . . . . . 9 𝑥𝐴 ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
75, 6bitr4i 186 . . . . . . . 8 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
87notbii 658 . . . . . . 7 𝑥 ∈ (V ∖ 𝐴) ↔ ¬ ¬ 𝑥𝐴)
92, 4, 83bitr2i 207 . . . . . 6 (𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ ¬ ¬ 𝑥𝐴)
109bibi1i 227 . . . . 5 ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴) ↔ (¬ ¬ 𝑥𝐴𝑥𝐴))
11 biimp 117 . . . . . 6 ((¬ ¬ 𝑥𝐴𝑥𝐴) → (¬ ¬ 𝑥𝐴𝑥𝐴))
12 id 19 . . . . . . 7 ((¬ ¬ 𝑥𝐴𝑥𝐴) → (¬ ¬ 𝑥𝐴𝑥𝐴))
13 notnot 619 . . . . . . 7 (𝑥𝐴 → ¬ ¬ 𝑥𝐴)
1412, 13impbid1 141 . . . . . 6 ((¬ ¬ 𝑥𝐴𝑥𝐴) → (¬ ¬ 𝑥𝐴𝑥𝐴))
1511, 14impbii 125 . . . . 5 ((¬ ¬ 𝑥𝐴𝑥𝐴) ↔ (¬ ¬ 𝑥𝐴𝑥𝐴))
1610, 15bitri 183 . . . 4 ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴) ↔ (¬ ¬ 𝑥𝐴𝑥𝐴))
17 df-stab 817 . . . 4 (STAB 𝑥𝐴 ↔ (¬ ¬ 𝑥𝐴𝑥𝐴))
1816, 17bitr4i 186 . . 3 ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴) ↔ STAB 𝑥𝐴)
1918albii 1450 . 2 (∀𝑥(𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴) ↔ ∀𝑥STAB 𝑥𝐴)
201, 19bitri 183 1 ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥STAB 𝑥𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  STAB wstab 816  ∀wal 1333   = wceq 1335   ∈ wcel 2128  Vcvv 2712   ∖ cdif 3099 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-stab 817  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104 This theorem is referenced by: (None)
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