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Theorem difun1 4304
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 4235 . . . 4 ((𝐴 ∩ (V ∖ 𝐵)) ∩ (V ∖ 𝐶)) = (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))
2 invdif 4284 . . . 4 ((𝐴 ∩ (V ∖ 𝐵)) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶)
31, 2eqtr3i 2764 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶)
4 undm 4302 . . . . 5 (V ∖ (𝐵𝐶)) = ((V ∖ 𝐵) ∩ (V ∖ 𝐶))
54ineq2i 4224 . . . 4 (𝐴 ∩ (V ∖ (𝐵𝐶))) = (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))
6 invdif 4284 . . . 4 (𝐴 ∩ (V ∖ (𝐵𝐶))) = (𝐴 ∖ (𝐵𝐶))
75, 6eqtr3i 2764 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = (𝐴 ∖ (𝐵𝐶))
83, 7eqtr3i 2764 . 2 ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶) = (𝐴 ∖ (𝐵𝐶))
9 invdif 4284 . . 3 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
109difeq1i 4131 . 2 ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶) = ((𝐴𝐵) ∖ 𝐶)
118, 10eqtr3i 2764 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  Vcvv 3477  cdif 3959  cun 3960  cin 3961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969
This theorem is referenced by:  dif32  4307  difabs  4308  difpr  4807  infdiffi  9695  mreexexlem4d  17691  nulmbl2  25584  unmbl  25585  caragenuncllem  46467
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