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Theorem difun1 4239
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 4171 . . . 4 ((𝐴 ∩ (V ∖ 𝐵)) ∩ (V ∖ 𝐶)) = (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))
2 invdif 4220 . . . 4 ((𝐴 ∩ (V ∖ 𝐵)) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶)
31, 2eqtr3i 2846 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶)
4 undm 4237 . . . . 5 (V ∖ (𝐵𝐶)) = ((V ∖ 𝐵) ∩ (V ∖ 𝐶))
54ineq2i 4161 . . . 4 (𝐴 ∩ (V ∖ (𝐵𝐶))) = (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))
6 invdif 4220 . . . 4 (𝐴 ∩ (V ∖ (𝐵𝐶))) = (𝐴 ∖ (𝐵𝐶))
75, 6eqtr3i 2846 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = (𝐴 ∖ (𝐵𝐶))
83, 7eqtr3i 2846 . 2 ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶) = (𝐴 ∖ (𝐵𝐶))
9 invdif 4220 . . 3 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
109difeq1i 4071 . 2 ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶) = ((𝐴𝐵) ∖ 𝐶)
118, 10eqtr3i 2846 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  Vcvv 3471  cdif 3907  cun 3908  cin 3909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917
This theorem is referenced by:  dif32  4242  difabs  4243  difpr  4709  infdiffi  9097  mreexexlem4d  16896  nulmbl2  24118  unmbl  24119  caragenuncllem  42942
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