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Theorem difun1 4288
Description: A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
Assertion
Ref Expression
difun1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)

Proof of Theorem difun1
StepHypRef Expression
1 inass 4218 . . . 4 ((𝐴 ∩ (V ∖ 𝐵)) ∩ (V ∖ 𝐶)) = (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))
2 invdif 4267 . . . 4 ((𝐴 ∩ (V ∖ 𝐵)) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶)
31, 2eqtr3i 2755 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶)
4 undm 4286 . . . . 5 (V ∖ (𝐵𝐶)) = ((V ∖ 𝐵) ∩ (V ∖ 𝐶))
54ineq2i 4207 . . . 4 (𝐴 ∩ (V ∖ (𝐵𝐶))) = (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))
6 invdif 4267 . . . 4 (𝐴 ∩ (V ∖ (𝐵𝐶))) = (𝐴 ∖ (𝐵𝐶))
75, 6eqtr3i 2755 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = (𝐴 ∖ (𝐵𝐶))
83, 7eqtr3i 2755 . 2 ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶) = (𝐴 ∖ (𝐵𝐶))
9 invdif 4267 . . 3 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
109difeq1i 4114 . 2 ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶) = ((𝐴𝐵) ∖ 𝐶)
118, 10eqtr3i 2755 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  Vcvv 3461  cdif 3941  cun 3942  cin 3943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951
This theorem is referenced by:  dif32  4291  difabs  4292  difpr  4808  infdiffi  9683  mreexexlem4d  17630  nulmbl2  25509  unmbl  25510  caragenuncllem  46038
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