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Theorem symdifv 5004
 Description: The symmetric difference with the universal class is the complement. (Contributed by Scott Fenton, 24-Apr-2012.)
Assertion
Ref Expression
symdifv (𝐴 △ V) = (V ∖ 𝐴)

Proof of Theorem symdifv
StepHypRef Expression
1 df-symdif 4222 . 2 (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴))
2 ssv 3994 . . . . 5 𝐴 ⊆ V
3 ssdif0 4326 . . . . 5 (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅)
42, 3mpbi 231 . . . 4 (𝐴 ∖ V) = ∅
54uneq1i 4138 . . 3 ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴))
6 uncom 4132 . . . 4 (∅ ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ ∅)
7 un0 4347 . . . 4 ((V ∖ 𝐴) ∪ ∅) = (V ∖ 𝐴)
86, 7eqtri 2848 . . 3 (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
95, 8eqtri 2848 . 2 ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
101, 9eqtri 2848 1 (𝐴 △ V) = (V ∖ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1530  Vcvv 3499   ∖ cdif 3936   ∪ cun 3937   ⊆ wss 3939   △ csymdif 4221  ∅c0 4294 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-symdif 4222  df-nul 4295 This theorem is referenced by: (None)
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