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Theorem compne 41160
 Description: The complement of 𝐴 is not equal to 𝐴. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 11-Nov-2021.)
Assertion
Ref Expression
compne (V ∖ 𝐴) ≠ 𝐴

Proof of Theorem compne
StepHypRef Expression
1 vn0 4254 . 2 V ≠ ∅
2 id 22 . . . . . . 7 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = 𝐴)
3 difeq1 4043 . . . . . . . 8 ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∖ 𝐴) = (𝐴𝐴))
4 difabs 4218 . . . . . . . 8 ((V ∖ 𝐴) ∖ 𝐴) = (V ∖ 𝐴)
5 difid 4284 . . . . . . . 8 (𝐴𝐴) = ∅
63, 4, 53eqtr3g 2856 . . . . . . 7 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = ∅)
72, 6eqtr3d 2835 . . . . . 6 ((V ∖ 𝐴) = 𝐴𝐴 = ∅)
87difeq2d 4050 . . . . 5 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = (V ∖ ∅))
9 dif0 4286 . . . . 5 (V ∖ ∅) = V
108, 9eqtrdi 2849 . . . 4 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = V)
1110, 6eqtr3d 2835 . . 3 ((V ∖ 𝐴) = 𝐴 → V = ∅)
1211necon3i 3019 . 2 (V ≠ ∅ → (V ∖ 𝐴) ≠ 𝐴)
131, 12ax-mp 5 1 (V ∖ 𝐴) ≠ 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ≠ wne 2987  Vcvv 3441   ∖ cdif 3878  ∅c0 4243 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244 This theorem is referenced by: (None)
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