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Theorem nincompl 4072
 Description: Anti-intersection with complement. (Contributed by SF, 2-Jan-2018.)
Assertion
Ref Expression
nincompl (A ⩃ ∼ A) = V

Proof of Theorem nincompl
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eqv 3565 . 2 ((A ⩃ ∼ A) = V ↔ x x (A ⩃ ∼ A))
2 pm3.24 852 . . 3 ¬ (x A ¬ x A)
3 vex 2862 . . . . 5 x V
43elnin 3224 . . . 4 (x (A ⩃ ∼ A) ↔ (x A x A))
53elcompl 3225 . . . . 5 (x A ↔ ¬ x A)
65nanbi2i 1299 . . . 4 ((x A x A) ↔ (x A ¬ x A))
7 df-nan 1288 . . . 4 ((x A ¬ x A) ↔ ¬ (x A ¬ x A))
84, 6, 73bitri 262 . . 3 (x (A ⩃ ∼ A) ↔ ¬ (x A ¬ x A))
92, 8mpbir 200 . 2 x (A ⩃ ∼ A)
101, 9mpgbir 1550 1 (A ⩃ ∼ A) = V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 358   ⊼ wnan 1287   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ⩃ cnin 3204   ∼ ccompl 3205 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212 This theorem is referenced by:  incompl  4073  uncompl  4074
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