Theorem disjdif2 4460

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 4100	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ ∅))
2		difin 4252	. 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
3		dif0 4358	. 2 ⊢ (𝐴 ∖ ∅) = 𝐴
4	1, 2, 3	3eqtr3g 2794	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∖ cdif 3928   ∩ cin 3930  ∅c0 4313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-in 3938  df-nul 4314

This theorem is referenced by:  undif5  4465  opwo0id  5477  setsfun0  17196  cnfldfun  21334  cnfldfunOLD  21347  ptbasfi  23524  sltlpss  27876  slelss  27877  fzdif2  32772  fzodif2  32773  chtvalz  34666  bj-2upln1upl  37047  disjresdif  38265  dvrelog2  42082  dvrelog3  42083  readvrec2  42371  readvrec  42372  gneispace  44125  dvmptfprodlem  45940