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Mathbox for Andrew Salmon |
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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 3977 | . 2 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴) | |
2 | 1 | eqabi 2874 | 1 ⊢ (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥 ∈ 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ∈ wcel 2105 {cab 2711 Vcvv 3477 ∖ cdif 3959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-dif 3965 |
This theorem is referenced by: (None) |
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