Theorem disjdif2 4436

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 4068	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ ∅))
2		difin 4218	. 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
3		dif0 4327	. 2 ⊢ (𝐴 ∖ ∅) = 𝐴
4	1, 2, 3	3eqtr3g 2819	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∖ cdif 3896   ∩ cin 3898  ∅c0 4279

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-in 3906  df-nul 4280

This theorem is used by:  undif5  4440  opwo0id  5469  setsfun0  17350  cnfldfun  21692  ptbasfi  23900  ltslpss  28294  leslss  28295  fzdif2  33382  fzodif2  33383  chtvalz  35258  bj-2upln1upl  37937  disjresdif  39177  dvrelog2  43114  dvrelog3  43115  readvrec2  43412  readvrec  43413  gneispace  45133  dvmptfprodlem  46953