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Theorem compne 42813
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 4302 . 2 V ≠ ∅
2 id 22 . . . . . . 7 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = 𝐴)
3 difeq1 4079 . . . . . . . 8 ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∖ 𝐴) = (𝐴𝐴))
4 difabs 4257 . . . . . . . 8 ((V ∖ 𝐴) ∖ 𝐴) = (V ∖ 𝐴)
5 difid 4334 . . . . . . . 8 (𝐴𝐴) = ∅
63, 4, 53eqtr3g 2796 . . . . . . 7 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = ∅)
72, 6eqtr3d 2775 . . . . . 6 ((V ∖ 𝐴) = 𝐴𝐴 = ∅)
87difeq2d 4086 . . . . 5 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = (V ∖ ∅))
9 dif0 4336 . . . . 5 (V ∖ ∅) = V
108, 9eqtrdi 2789 . . . 4 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = V)
1110, 6eqtr3d 2775 . . 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 1542  wne 2940  Vcvv 3447  cdif 3911  c0 4286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-nul 4287
This theorem is referenced by: (None)
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