Theorem incompl 4074

Description: Intersection with complement. (Contributed by SF, 2-Jan-2018.)

Assertion

Ref	Expression
incompl	|- (A i^i ∼ A) = (/)

Proof of Theorem incompl

Step	Hyp	Ref	Expression
1		df-in 3214	. 2 |- (A i^i ∼ A) = ∼ (A &ncap ∼ A)
2		nincompl 4073	. . 3 |- (A &ncap ∼ A) = _V
3	2	compleqi 3245	. 2 |- ∼ (A &ncap ∼ A) = ∼ _V
4		complV 4071	. 2 |- ∼ _V = (/)
5	1, 3, 4	3eqtri 2377	1 |- (A i^i ∼ A) = (/)

Syntax hints:   = wceq 1642  _Vcvv 2860   &ncap cnin 3205   ∼ ccompl 3206   i^i cin 3209  (/)c0 3551

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  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  inindif  4076  nnsucelrlem3  4427  vfintle  4547  vfin1cltv  4548  fnfullfun  5859  fvfullfun  5865  sbthlem1  6204