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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 4303 | . 2 ⊢ V ≠ ∅ | |
2 | id 22 | . . . . . . 7 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = 𝐴) | |
3 | difeq1 4091 | . . . . . . . 8 ⊢ ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∖ 𝐴) = (𝐴 ∖ 𝐴)) | |
4 | difabs 4267 | . . . . . . . 8 ⊢ ((V ∖ 𝐴) ∖ 𝐴) = (V ∖ 𝐴) | |
5 | difid 4329 | . . . . . . . 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 4098 | . . . . 5 ⊢ ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = (V ∖ ∅)) |
9 | dif0 4331 | . . . . 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 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ≠ wne 3016 Vcvv 3494 ∖ cdif 3932 ∅c0 4290 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 |
This theorem is referenced by: (None) |
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