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Theorem symdifv 4589
Description: Symmetric difference with the universal class. (Contributed by Scott Fenton, 24-Apr-2012.)
Assertion
Ref Expression
symdifv (𝐴 △ V) = (V ∖ 𝐴)

Proof of Theorem symdifv
StepHypRef Expression
1 df-symdif 3836 . 2 (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴))
2 ssv 3617 . . . . 5 𝐴 ⊆ V
3 ssdif0 3933 . . . . 5 (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅)
42, 3mpbi 220 . . . 4 (𝐴 ∖ V) = ∅
54uneq1i 3755 . . 3 ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴))
6 uncom 3749 . . . 4 (∅ ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ ∅)
7 un0 3958 . . . 4 ((V ∖ 𝐴) ∪ ∅) = (V ∖ 𝐴)
86, 7eqtri 2642 . . 3 (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
95, 8eqtri 2642 . 2 ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
101, 9eqtri 2642 1 (𝐴 △ V) = (V ∖ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1481  Vcvv 3195  cdif 3564  cun 3565  wss 3567  csymdif 3835  c0 3907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-symdif 3836  df-nul 3908
This theorem is referenced by: (None)
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