Theorem compne 40793

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 4304	. 2 ⊢ V ≠ ∅
2		id 22	. . . . . . 7 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = 𝐴)
3		difeq1 4092	. . . . . . . 8 ⊢ ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∖ 𝐴) = (𝐴 ∖ 𝐴))
4		difabs 4268	. . . . . . . 8 ⊢ ((V ∖ 𝐴) ∖ 𝐴) = (V ∖ 𝐴)
5		difid 4330	. . . . . . . 8 ⊢ (𝐴 ∖ 𝐴) = ∅
6	3, 4, 5	3eqtr3g 2879	. . . . . . 7 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = ∅)
7	2, 6	eqtr3d 2858	. . . . . 6 ⊢ ((V ∖ 𝐴) = 𝐴 → 𝐴 = ∅)
8	7	difeq2d 4099	. . . . 5 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = (V ∖ ∅))
9		dif0 4332	. . . . 5 ⊢ (V ∖ ∅) = V
10	8, 9	syl6eq 2872	. . . 4 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = V)
11	10, 6	eqtr3d 2858	. . 3 ⊢ ((V ∖ 𝐴) = 𝐴 → V = ∅)
12	11	necon3i 3048	. 2 ⊢ (V ≠ ∅ → (V ∖ 𝐴) ≠ 𝐴)
13	1, 12	ax-mp 5	1 ⊢ (V ∖ 𝐴) ≠ 𝐴

Syntax hints:   = wceq 1537   ≠ wne 3016  Vcvv 3494   ∖ cdif 3933  ∅c0 4291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292

This theorem is referenced by: (None)