ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ddifstab GIF version

Theorem ddifstab 3176
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 2109 . 2 ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥(𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴))
2 eldif 3048 . . . . . . 7 (𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
3 vex 2661 . . . . . . . 8 𝑥 ∈ V
43biantrur 299 . . . . . . 7 𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
5 eldif 3048 . . . . . . . . 9 (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
63biantrur 299 . . . . . . . . 9 𝑥𝐴 ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
75, 6bitr4i 186 . . . . . . . 8 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
87notbii 640 . . . . . . 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 bi1 117 . . . . . 6 ((¬ ¬ 𝑥𝐴𝑥𝐴) → (¬ ¬ 𝑥𝐴𝑥𝐴))
12 id 19 . . . . . . 7 ((¬ ¬ 𝑥𝐴𝑥𝐴) → (¬ ¬ 𝑥𝐴𝑥𝐴))
13 notnot 601 . . . . . . 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 799 . . . 4 (STAB 𝑥𝐴 ↔ (¬ ¬ 𝑥𝐴𝑥𝐴))
1816, 17bitr4i 186 . . 3 ((𝑥 ∈ (V ∖ (V ∖ 𝐴)) ↔ 𝑥𝐴) ↔ STAB 𝑥𝐴)
1918albii 1429 . 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 798  wal 1312   = wceq 1314  wcel 1463  Vcvv 2658  cdif 3036
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-stab 799  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator