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Theorem symdifv 4525
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 3802 . 2 (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴))
2 ssv 3584 . . . . 5 𝐴 ⊆ V
3 ssdif0 3892 . . . . 5 (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅)
42, 3mpbi 218 . . . 4 (𝐴 ∖ V) = ∅
54uneq1i 3721 . . 3 ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴))
6 uncom 3715 . . . 4 (∅ ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ ∅)
7 un0 3915 . . . 4 ((V ∖ 𝐴) ∪ ∅) = (V ∖ 𝐴)
86, 7eqtri 2628 . . 3 (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
95, 8eqtri 2628 . 2 ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
101, 9eqtri 2628 1 (𝐴 △ V) = (V ∖ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  Vcvv 3169  cdif 3533  cun 3534  wss 3536  csymdif 3801  c0 3870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-symdif 3802  df-nul 3871
This theorem is referenced by: (None)
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