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

Theorem difdif2 3865
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 3862 . 2 (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
2 invdif 3849 . . . 4 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
32eqcomi 2635 . . 3 (𝐵𝐶) = (𝐵 ∩ (V ∖ 𝐶))
43difeq2i 3708 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
5 dfin2 3843 . . 3 (𝐴𝐶) = (𝐴 ∖ (V ∖ 𝐶))
65uneq2i 3747 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
71, 4, 63eqtr4i 2658 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  Vcvv 3191  cdif 3557  cun 3558  cin 3559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567
This theorem is referenced by:  restmetu  22280  difelcarsg  30145  mblfinlem3  33066  mblfinlem4  33067
  Copyright terms: Public domain W3C validator