Theorem disjdif2 4503

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 4143	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ ∅))
2		difin 4291	. 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
3		dif0 4400	. 2 ⊢ (𝐴 ∖ ∅) = 𝐴
4	1, 2, 3	3eqtr3g 2803	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴)

Syntax hints:   → wi 4   = wceq 1537   ∖ cdif 3973   ∩ cin 3975  ∅c0 4352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-in 3983  df-nul 4353

This theorem is referenced by:  undif5  4508  opwo0id  5516  setsfun0  17219  cnfldfun  21401  cnfldfunOLD  21414  ptbasfi  23610  sltlpss  27963  slelss  27964  fzdif2  32796  fzodif2  32797  chtvalz  34606  bj-2upln1upl  36990  disjresdif  38197  dvrelog2  42021  dvrelog3  42022  gneispace  44096  dvmptfprodlem  45865