Theorem disjdif2 4446

Description: The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.)

Assertion

Ref	Expression
disjdif2	⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴)

Proof of Theorem disjdif2

Step	Hyp	Ref	Expression
1		difeq2 4083	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ ∅))
2		difin 4233	. 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
3		dif0 4341	. 2 ⊢ (𝐴 ∖ ∅) = 𝐴
4	1, 2, 3	3eqtr3g 2827	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴)

Syntax hints:   → wi 4   = wceq 1567   ∖ cdif 3910   ∩ cin 3912  ∅c0 4294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-in 3920  df-nul 4295

This theorem is referenced by:  undif5  4450  opwo0id  5481  setsfun0  17232  cnfldfun  21505  ptbasfi  23707  ltslpss  28067  leslss  28068  fzdif2  33076  fzodif2  33077  chtvalz  34961  bj-2upln1upl  37548  disjresdif  38784  dvrelog2  42721  dvrelog3  42722  readvrec2  43012  readvrec  43013  gneispace  44752  dvmptfprodlem  46550