Theorem disjdif2 4425

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 4062	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ ∅))
2		difin 4207	. 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
3		dif0 4318	. 2 ⊢ (𝐴 ∖ ∅) = 𝐴
4	1, 2, 3	3eqtr3g 2799	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∖ cdif 3894   ∩ cin 3896  ∅c0 4268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-dif 3900  df-in 3904  df-nul 4269

This theorem is referenced by:  undif5  4428  opwo0id  5435  setsfun0  16962  cnfldfun  20707  ptbasfi  22830  fzdif2  31340  fzodif2  31341  chtvalz  32850  sltlpss  34178  bj-2upln1upl  35303  disjresdif  36497  dvrelog2  40319  dvrelog3  40320  gneispace  42054  dvmptfprodlem  43810