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Mirrors > Home > MPE Home > Th. List > dfsymdif2 | Structured version Visualization version GIF version |
Description: Alternate definition of the symmetric difference. (Contributed by BJ, 30-Apr-2020.) |
Ref | Expression |
---|---|
dfsymdif2 | ⊢ (𝐴 △ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsymdifxor 4048 | . 2 ⊢ (𝑥 ∈ (𝐴 △ 𝐵) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)) | |
2 | 1 | abbi2i 2915 | 1 ⊢ (𝐴 △ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)} |
Colors of variables: wff setvar class |
Syntax hints: ⊻ wxo 1634 = wceq 1653 ∈ wcel 2157 {cab 2785 △ csymdif 4040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-xor 1635 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-v 3387 df-dif 3772 df-un 3774 df-symdif 4041 |
This theorem is referenced by: (None) |
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