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Theorem dfin3 4096
 Description: Intersection defined in terms of union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
Assertion
Ref Expression
dfin3 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))

Proof of Theorem dfin3
StepHypRef Expression
1 ddif 3969 . 2 (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵)))) = (𝐴 ∖ (V ∖ 𝐵))
2 dfun2 4089 . . . 4 ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵)))
3 ddif 3969 . . . . . 6 (V ∖ (V ∖ 𝐴)) = 𝐴
43difeq1i 3951 . . . . 5 ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ 𝐵))
54difeq2i 3952 . . . 4 (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵))) = (V ∖ (𝐴 ∖ (V ∖ 𝐵)))
62, 5eqtri 2849 . . 3 ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ (𝐴 ∖ (V ∖ 𝐵)))
76difeq2i 3952 . 2 (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) = (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵))))
8 dfin2 4090 . 2 (𝐴𝐵) = (𝐴 ∖ (V ∖ 𝐵))
91, 7, 83eqtr4ri 2860 1 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))
 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:  difindi  4111
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