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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 4345 | . 2 ⊢ V ≠ ∅ | |
| 2 | id 22 | . . . . . . 7 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = 𝐴) | |
| 3 | difeq1 4119 | . . . . . . . 8 ⊢ ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∖ 𝐴) = (𝐴 ∖ 𝐴)) | |
| 4 | difabs 4303 | . . . . . . . 8 ⊢ ((V ∖ 𝐴) ∖ 𝐴) = (V ∖ 𝐴) | |
| 5 | difid 4376 | . . . . . . . 8 ⊢ (𝐴 ∖ 𝐴) = ∅ | |
| 6 | 3, 4, 5 | 3eqtr3g 2800 | . . . . . . 7 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = ∅) |
| 7 | 2, 6 | eqtr3d 2779 | . . . . . 6 ⊢ ((V ∖ 𝐴) = 𝐴 → 𝐴 = ∅) |
| 8 | 7 | difeq2d 4126 | . . . . 5 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = (V ∖ ∅)) |
| 9 | dif0 4378 | . . . . 5 ⊢ (V ∖ ∅) = V | |
| 10 | 8, 9 | eqtrdi 2793 | . . . 4 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = V) |
| 11 | 10, 6 | eqtr3d 2779 | . . 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 1540 ≠ wne 2940 Vcvv 3480 ∖ cdif 3948 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-nul 4334 |
| This theorem is referenced by: (None) |
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