Theorem nincompl 4073

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.

Step	Hyp	Ref	Expression
1		eqv 3566	. 2 ⊢ ((A ⩃ ∼ A) = V ↔ ∀x x ∈ (A ⩃ ∼ A))
2		pm3.24 852	. . 3 ⊢ ¬ (x ∈ A ∧ ¬ x ∈ A)
3		vex 2863	. . . . 5 ⊢ x ∈ V
4	3	elnin 3225	. . . 4 ⊢ (x ∈ (A ⩃ ∼ A) ↔ (x ∈ A ⊼ x ∈ ∼ A))
5	3	elcompl 3226	. . . . 5 ⊢ (x ∈ ∼ A ↔ ¬ x ∈ A)
6	5	nanbi2i 1299	. . . 4 ⊢ ((x ∈ A ⊼ x ∈ ∼ A) ↔ (x ∈ A ⊼ ¬ x ∈ A))
7		df-nan 1288	. . . 4 ⊢ ((x ∈ A ⊼ ¬ x ∈ A) ↔ ¬ (x ∈ A ∧ ¬ x ∈ A))
8	4, 6, 7	3bitri 262	. . 3 ⊢ (x ∈ (A ⩃ ∼ A) ↔ ¬ (x ∈ A ∧ ¬ x ∈ A))
9	2, 8	mpbir 200	. 2 ⊢ x ∈ (A ⩃ ∼ A)
10	1, 9	mpgbir 1550	1 ⊢ (A ⩃ ∼ A) = V

Syntax hints:  ¬ wn 3   ∧ wa 358   ⊼ wnan 1287   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ⩃ cnin 3205   ∼ ccompl 3206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-v 2862  df-nin 3212  df-compl 3213

This theorem is referenced by:  incompl  4074  uncompl  4075