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Theorem difun1 4117
 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 4048 . . . 4 ((𝐴 ∩ (V ∖ 𝐵)) ∩ (V ∖ 𝐶)) = (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))
2 invdif 4098 . . . 4 ((𝐴 ∩ (V ∖ 𝐵)) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶)
31, 2eqtr3i 2851 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶)
4 undm 4115 . . . . 5 (V ∖ (𝐵𝐶)) = ((V ∖ 𝐵) ∩ (V ∖ 𝐶))
54ineq2i 4038 . . . 4 (𝐴 ∩ (V ∖ (𝐵𝐶))) = (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))
6 invdif 4098 . . . 4 (𝐴 ∩ (V ∖ (𝐵𝐶))) = (𝐴 ∖ (𝐵𝐶))
75, 6eqtr3i 2851 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = (𝐴 ∖ (𝐵𝐶))
83, 7eqtr3i 2851 . 2 ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶) = (𝐴 ∖ (𝐵𝐶))
9 invdif 4098 . . 3 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
109difeq1i 3951 . 2 ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶) = ((𝐴𝐵) ∖ 𝐶)
118, 10eqtr3i 2851 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1658  Vcvv 3414   ∖ cdif 3795   ∪ cun 3796   ∩ cin 3797 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805 This theorem is referenced by:  dif32  4120  difabs  4121  difpr  4552  infdiffi  8832  mreexexlem4d  16660  nulmbl2  23702  unmbl  23703  caragenuncllem  41520
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