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Theorem compne 41732
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 4253 . 2 V ≠ ∅
2 id 22 . . . . . . 7 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = 𝐴)
3 difeq1 4030 . . . . . . . 8 ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∖ 𝐴) = (𝐴𝐴))
4 difabs 4208 . . . . . . . 8 ((V ∖ 𝐴) ∖ 𝐴) = (V ∖ 𝐴)
5 difid 4285 . . . . . . . 8 (𝐴𝐴) = ∅
63, 4, 53eqtr3g 2801 . . . . . . 7 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = ∅)
72, 6eqtr3d 2779 . . . . . 6 ((V ∖ 𝐴) = 𝐴𝐴 = ∅)
87difeq2d 4037 . . . . 5 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = (V ∖ ∅))
9 dif0 4287 . . . . 5 (V ∖ ∅) = V
108, 9eqtrdi 2794 . . . 4 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = V)
1110, 6eqtr3d 2779 . . 3 ((V ∖ 𝐴) = 𝐴 → V = ∅)
1211necon3i 2973 . 2 (V ≠ ∅ → (V ∖ 𝐴) ≠ 𝐴)
131, 12ax-mp 5 1 (V ∖ 𝐴) ≠ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  wne 2940  Vcvv 3408  cdif 3863  c0 4237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-nul 4238
This theorem is referenced by: (None)
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