MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difdif2 Structured version   Visualization version   GIF version

Theorem difdif2 4296
Description: Class difference by a class difference. (Contributed by Thierry Arnoux, 18-Dec-2017.)
Assertion
Ref Expression
difdif2 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Proof of Theorem difdif2
StepHypRef Expression
1 difindi 4292 . 2 (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
2 invdif 4279 . . . 4 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
32eqcomi 2746 . . 3 (𝐵𝐶) = (𝐵 ∩ (V ∖ 𝐶))
43difeq2i 4123 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
5 dfin2 4271 . . 3 (𝐴𝐶) = (𝐴 ∖ (V ∖ 𝐶))
65uneq2i 4165 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
71, 4, 63eqtr4i 2775 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  Vcvv 3480  cdif 3948  cun 3949  cin 3950
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958
This theorem is referenced by:  psdmullem  22169  restmetu  24583  difelcarsg  34312  mblfinlem3  37666  mblfinlem4  37667
  Copyright terms: Public domain W3C validator