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Theorem compne 39419
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 4126 . 2 V ≠ ∅
2 id 22 . . . . . . 7 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = 𝐴)
3 difeq1 3920 . . . . . . . 8 ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∖ 𝐴) = (𝐴𝐴))
4 difabs 4093 . . . . . . . 8 ((V ∖ 𝐴) ∖ 𝐴) = (V ∖ 𝐴)
5 difid 4150 . . . . . . . 8 (𝐴𝐴) = ∅
63, 4, 53eqtr3g 2857 . . . . . . 7 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = ∅)
72, 6eqtr3d 2836 . . . . . 6 ((V ∖ 𝐴) = 𝐴𝐴 = ∅)
87difeq2d 3927 . . . . 5 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = (V ∖ ∅))
9 dif0 4152 . . . . 5 (V ∖ ∅) = V
108, 9syl6eq 2850 . . . 4 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = V)
1110, 6eqtr3d 2836 . . 3 ((V ∖ 𝐴) = 𝐴 → V = ∅)
1211necon3i 3004 . 2 (V ≠ ∅ → (V ∖ 𝐴) ≠ 𝐴)
131, 12ax-mp 5 1 (V ∖ 𝐴) ≠ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  wne 2972  Vcvv 3386  cdif 3767  c0 4116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117
This theorem is referenced by: (None)
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