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

Step	Hyp	Ref	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) = (𝐴 ∖ 𝐴))
4		difabs 4093	. . . . . . . 8 ⊢ ((V ∖ 𝐴) ∖ 𝐴) = (V ∖ 𝐴)
5		difid 4150	. . . . . . . 8 ⊢ (𝐴 ∖ 𝐴) = ∅
6	3, 4, 5	3eqtr3g 2857	. . . . . . 7 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = ∅)
7	2, 6	eqtr3d 2836	. . . . . 6 ⊢ ((V ∖ 𝐴) = 𝐴 → 𝐴 = ∅)
8	7	difeq2d 3927	. . . . 5 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = (V ∖ ∅))
9		dif0 4152	. . . . 5 ⊢ (V ∖ ∅) = V
10	8, 9	syl6eq 2850	. . . 4 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = V)
11	10, 6	eqtr3d 2836	. . 3 ⊢ ((V ∖ 𝐴) = 𝐴 → V = ∅)
12	11	necon3i 3004	. 2 ⊢ (V ≠ ∅ → (V ∖ 𝐴) ≠ 𝐴)
13	1, 12	ax-mp 5	1 ⊢ (V ∖ 𝐴) ≠ 𝐴

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)