Theorem incompl 4074

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

Assertion

Ref	Expression
incompl	⊢ (A ∩ ∼ A) = ∅

Proof of Theorem incompl

Step	Hyp	Ref	Expression
1		df-in 3214	. 2 ⊢ (A ∩ ∼ A) = ∼ (A ⩃ ∼ A)
2		nincompl 4073	. . 3 ⊢ (A ⩃ ∼ A) = V
3	2	compleqi 3245	. 2 ⊢ ∼ (A ⩃ ∼ A) = ∼ V
4		complV 4071	. 2 ⊢ ∼ V = ∅
5	1, 3, 4	3eqtri 2377	1 ⊢ (A ∩ ∼ A) = ∅

Syntax hints:   = wceq 1642  Vcvv 2860   ⩃ cnin 3205   ∼ ccompl 3206   ∩ 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