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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 3948 | . 2 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴) | |
2 | 1 | abbi2i 2950 | 1 ⊢ (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥 ∈ 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∈ wcel 2105 {cab 2796 Vcvv 3492 ∖ cdif 3930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-dif 3936 |
This theorem is referenced by: (None) |
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