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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 4351 | . 2 ⊢ V ≠ ∅ | |
2 | id 22 | . . . . . . 7 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = 𝐴) | |
3 | difeq1 4129 | . . . . . . . 8 ⊢ ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∖ 𝐴) = (𝐴 ∖ 𝐴)) | |
4 | difabs 4309 | . . . . . . . 8 ⊢ ((V ∖ 𝐴) ∖ 𝐴) = (V ∖ 𝐴) | |
5 | difid 4382 | . . . . . . . 8 ⊢ (𝐴 ∖ 𝐴) = ∅ | |
6 | 3, 4, 5 | 3eqtr3g 2798 | . . . . . . 7 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = ∅) |
7 | 2, 6 | eqtr3d 2777 | . . . . . 6 ⊢ ((V ∖ 𝐴) = 𝐴 → 𝐴 = ∅) |
8 | 7 | difeq2d 4136 | . . . . 5 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = (V ∖ ∅)) |
9 | dif0 4384 | . . . . 5 ⊢ (V ∖ ∅) = V | |
10 | 8, 9 | eqtrdi 2791 | . . . 4 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = V) |
11 | 10, 6 | eqtr3d 2777 | . . 3 ⊢ ((V ∖ 𝐴) = 𝐴 → V = ∅) |
12 | 11 | necon3i 2971 | . 2 ⊢ (V ≠ ∅ → (V ∖ 𝐴) ≠ 𝐴) |
13 | 1, 12 | ax-mp 5 | 1 ⊢ (V ∖ 𝐴) ≠ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ≠ wne 2938 Vcvv 3478 ∖ cdif 3960 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-nul 4340 |
This theorem is referenced by: (None) |
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