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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 4150 | . 2 ⊢ (𝑥 ∈ (𝐴 △ 𝐵) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)) | |
2 | 1 | abbi2i 2869 | 1 ⊢ (𝐴 △ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)} |
Colors of variables: wff setvar class |
Syntax hints: ⊻ wxo 1507 = wceq 1543 ∈ wcel 2112 {cab 2714 △ csymdif 4142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-xor 1508 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-dif 3856 df-un 3858 df-symdif 4143 |
This theorem is referenced by: (None) |
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