Theorem compne 43190

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

Step	Hyp	Ref	Expression
1		vn0 4338	. 2 ⊢ V ≠ ∅
2		id 22	. . . . . . 7 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = 𝐴)
3		difeq1 4115	. . . . . . . 8 ⊢ ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∖ 𝐴) = (𝐴 ∖ 𝐴))
4		difabs 4293	. . . . . . . 8 ⊢ ((V ∖ 𝐴) ∖ 𝐴) = (V ∖ 𝐴)
5		difid 4370	. . . . . . . 8 ⊢ (𝐴 ∖ 𝐴) = ∅
6	3, 4, 5	3eqtr3g 2795	. . . . . . 7 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = ∅)
7	2, 6	eqtr3d 2774	. . . . . 6 ⊢ ((V ∖ 𝐴) = 𝐴 → 𝐴 = ∅)
8	7	difeq2d 4122	. . . . 5 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = (V ∖ ∅))
9		dif0 4372	. . . . 5 ⊢ (V ∖ ∅) = V
10	8, 9	eqtrdi 2788	. . . 4 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = V)
11	10, 6	eqtr3d 2774	. . 3 ⊢ ((V ∖ 𝐴) = 𝐴 → V = ∅)
12	11	necon3i 2973	. 2 ⊢ (V ≠ ∅ → (V ∖ 𝐴) ≠ 𝐴)
13	1, 12	ax-mp 5	1 ⊢ (V ∖ 𝐴) ≠ 𝐴

Syntax hints:   = wceq 1541   ≠ wne 2940  Vcvv 3474   ∖ cdif 3945  ∅c0 4322

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-nul 4323

This theorem is referenced by: (None)