Theorem compne 38643

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 3924	. 2 ⊢ V ≠ ∅
2		id 22	. . . . . . 7 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = 𝐴)
3		difeq1 3721	. . . . . . . 8 ⊢ ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∖ 𝐴) = (𝐴 ∖ 𝐴))
4		difabs 3892	. . . . . . . 8 ⊢ ((V ∖ 𝐴) ∖ 𝐴) = (V ∖ 𝐴)
5		difid 3948	. . . . . . . 8 ⊢ (𝐴 ∖ 𝐴) = ∅
6	3, 4, 5	3eqtr3g 2679	. . . . . . 7 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = ∅)
7	2, 6	eqtr3d 2658	. . . . . 6 ⊢ ((V ∖ 𝐴) = 𝐴 → 𝐴 = ∅)
8	7	difeq2d 3728	. . . . 5 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = (V ∖ ∅))
9		dif0 3950	. . . . 5 ⊢ (V ∖ ∅) = V
10	8, 9	syl6eq 2672	. . . 4 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = V)
11	10, 6	eqtr3d 2658	. . 3 ⊢ ((V ∖ 𝐴) = 𝐴 → V = ∅)
12	11	necon3i 2826	. 2 ⊢ (V ≠ ∅ → (V ∖ 𝐴) ≠ 𝐴)
13	1, 12	ax-mp 5	1 ⊢ (V ∖ 𝐴) ≠ 𝐴

Syntax hints:   = wceq 1483   ≠ wne 2794  Vcvv 3200   ∖ cdif 3571  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by: (None)