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| Mirrors > Home > MPE Home > Th. List > Mathboxes > compeq | Structured version Visualization version GIF version | ||
| Description: Equality between two ways of saying "the complement of 𝐴". (Contributed by Andrew Salmon, 15-Jul-2011.) |
| Ref | Expression |
|---|---|
| compeq | ⊢ (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥 ∈ 𝐴} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | velcomp 3966 | . 2 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴) | |
| 2 | 1 | eqabi 2877 | 1 ⊢ (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥 ∈ 𝐴} |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 {cab 2714 Vcvv 3480 ∖ cdif 3948 |
| 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 |
| This theorem is referenced by: (None) |
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