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Theorem compne 37665
 Description: The complement of 𝐴 is not equal to 𝐴. (Contributed by Andrew Salmon, 15-Jul-2011.)
Assertion
Ref Expression
compne (V ∖ 𝐴) ≠ 𝐴

Proof of Theorem compne
StepHypRef Expression
1 vn0 3883 . 2 V ≠ ∅
2 ssun1 3738 . . . . . . . 8 V ⊆ (V ∪ 𝐴)
3 ssv 3588 . . . . . . . 8 (V ∪ 𝐴) ⊆ V
42, 3eqssi 3584 . . . . . . 7 V = (V ∪ 𝐴)
5 undif1 3995 . . . . . . 7 ((V ∖ 𝐴) ∪ 𝐴) = (V ∪ 𝐴)
64, 5eqtr4i 2635 . . . . . 6 V = ((V ∖ 𝐴) ∪ 𝐴)
7 uneq1 3722 . . . . . 6 ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∪ 𝐴) = (𝐴𝐴))
86, 7syl5eq 2656 . . . . 5 ((V ∖ 𝐴) = 𝐴 → V = (𝐴𝐴))
9 unidm 3718 . . . . 5 (𝐴𝐴) = 𝐴
108, 9syl6eq 2660 . . . 4 ((V ∖ 𝐴) = 𝐴 → V = 𝐴)
11 difabs 3851 . . . . . . 7 ((V ∖ 𝐴) ∖ 𝐴) = (V ∖ 𝐴)
12 id 22 . . . . . . 7 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = 𝐴)
1311, 12syl5req 2657 . . . . . 6 ((V ∖ 𝐴) = 𝐴𝐴 = ((V ∖ 𝐴) ∖ 𝐴))
14 difeq1 3683 . . . . . 6 ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∖ 𝐴) = (𝐴𝐴))
1513, 14eqtrd 2644 . . . . 5 ((V ∖ 𝐴) = 𝐴𝐴 = (𝐴𝐴))
16 difid 3902 . . . . 5 (𝐴𝐴) = ∅
1715, 16syl6eq 2660 . . . 4 ((V ∖ 𝐴) = 𝐴𝐴 = ∅)
1810, 17eqtrd 2644 . . 3 ((V ∖ 𝐴) = 𝐴 → V = ∅)
1918necon3i 2814 . 2 (V ≠ ∅ → (V ∖ 𝐴) ≠ 𝐴)
201, 19ax-mp 5 1 (V ∖ 𝐴) ≠ 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ≠ wne 2780  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875 This theorem is referenced by: (None)
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