![]() |
Mathbox for Andrew Salmon |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > compne | Structured version Visualization version GIF version |
Description: The complement of 𝐴 is not equal to 𝐴. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 11-Nov-2021.) |
Ref | Expression |
---|---|
compne | ⊢ (V ∖ 𝐴) ≠ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vn0 4302 | . 2 ⊢ V ≠ ∅ | |
2 | id 22 | . . . . . . 7 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = 𝐴) | |
3 | difeq1 4079 | . . . . . . . 8 ⊢ ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∖ 𝐴) = (𝐴 ∖ 𝐴)) | |
4 | difabs 4257 | . . . . . . . 8 ⊢ ((V ∖ 𝐴) ∖ 𝐴) = (V ∖ 𝐴) | |
5 | difid 4334 | . . . . . . . 8 ⊢ (𝐴 ∖ 𝐴) = ∅ | |
6 | 3, 4, 5 | 3eqtr3g 2796 | . . . . . . 7 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = ∅) |
7 | 2, 6 | eqtr3d 2775 | . . . . . 6 ⊢ ((V ∖ 𝐴) = 𝐴 → 𝐴 = ∅) |
8 | 7 | difeq2d 4086 | . . . . 5 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = (V ∖ ∅)) |
9 | dif0 4336 | . . . . 5 ⊢ (V ∖ ∅) = V | |
10 | 8, 9 | eqtrdi 2789 | . . . 4 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = V) |
11 | 10, 6 | eqtr3d 2775 | . . 3 ⊢ ((V ∖ 𝐴) = 𝐴 → V = ∅) |
12 | 11 | necon3i 2973 | . 2 ⊢ (V ≠ ∅ → (V ∖ 𝐴) ≠ 𝐴) |
13 | 1, 12 | ax-mp 5 | 1 ⊢ (V ∖ 𝐴) ≠ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ≠ wne 2940 Vcvv 3447 ∖ cdif 3911 ∅c0 4286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-nul 4287 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |