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Theorem symdifv 4730
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 3991 . 2 (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴))
2 ssv 3772 . . . . 5 𝐴 ⊆ V
3 ssdif0 4087 . . . . 5 (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅)
42, 3mpbi 220 . . . 4 (𝐴 ∖ V) = ∅
54uneq1i 3912 . . 3 ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴))
6 uncom 3906 . . . 4 (∅ ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ ∅)
7 un0 4109 . . . 4 ((V ∖ 𝐴) ∪ ∅) = (V ∖ 𝐴)
86, 7eqtri 2792 . . 3 (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
95, 8eqtri 2792 . 2 ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
101, 9eqtri 2792 1 (𝐴 △ V) = (V ∖ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1630  Vcvv 3349  cdif 3718  cun 3719  wss 3721  csymdif 3990  c0 4061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-symdif 3991  df-nul 4062
This theorem is referenced by: (None)
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