Theorem incompl 4073

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 3213	. 2 ⊢ (A ∩ ∼ A) = ∼ (A ⩃ ∼ A)
2		nincompl 4072	. . 3 ⊢ (A ⩃ ∼ A) = V
3	2	compleqi 3244	. 2 ⊢ ∼ (A ⩃ ∼ A) = ∼ V
4		complV 4070	. 2 ⊢ ∼ V = ∅
5	1, 3, 4	3eqtri 2377	1 ⊢ (A ∩ ∼ A) = ∅

Syntax hints:   = wceq 1642  Vcvv 2859   ⩃ cnin 3204   ∼ ccompl 3205   ∩ cin 3208  ∅c0 3550

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  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551

This theorem is referenced by:  inindif  4075  nnsucelrlem3  4426  vfintle  4546  vfin1cltv  4547  fnfullfun  5858  fvfullfun  5864  sbthlem1  6203