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Theorem symdifv 4783
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 4036 . 2 (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴))
2 ssv 3816 . . . . 5 𝐴 ⊆ V
3 ssdif0 4137 . . . . 5 (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅)
42, 3mpbi 221 . . . 4 (𝐴 ∖ V) = ∅
54uneq1i 3956 . . 3 ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴))
6 uncom 3950 . . . 4 (∅ ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ ∅)
7 un0 4159 . . . 4 ((V ∖ 𝐴) ∪ ∅) = (V ∖ 𝐴)
86, 7eqtri 2824 . . 3 (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
95, 8eqtri 2824 . 2 ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
101, 9eqtri 2824 1 (𝐴 △ V) = (V ∖ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1637  Vcvv 3387  cdif 3760  cun 3761  wss 3763  csymdif 4035  c0 4110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-v 3389  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-symdif 4036  df-nul 4111
This theorem is referenced by: (None)
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