Theorem compldif 4070

Description: Complement in terms of difference. (Contributed by SF, 2-Jan-2018.)

Assertion

Ref	Expression
compldif	⊢ ∼ A = (V ∖ A)

Proof of Theorem compldif

Step	Hyp	Ref	Expression
1		df-dif 3216	. 2 ⊢ (V ∖ A) = (V ∩ ∼ A)
2		incom 3449	. 2 ⊢ (V ∩ ∼ A) = ( ∼ A ∩ V)
3		inv1 3578	. 2 ⊢ ( ∼ A ∩ V) = ∼ A
4	1, 2, 3	3eqtrri 2378	1 ⊢ ∼ A = (V ∖ A)

Syntax hints:   = wceq 1642  Vcvv 2860   ∼ ccompl 3206   ∖ cdif 3207   ∩ cin 3209

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

This theorem is referenced by:  complV  4071