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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 4244 | . 2 ⊢ (𝑥 ∈ (𝐴 △ 𝐵) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)) | |
2 | 1 | eqabi 2863 | 1 ⊢ (𝐴 △ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)} |
Colors of variables: wff setvar class |
Syntax hints: ⊻ wxo 1504 = wceq 1533 ∈ wcel 2098 {cab 2703 △ csymdif 4236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-xor 1505 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-dif 3946 df-un 3948 df-symdif 4237 |
This theorem is referenced by: (None) |
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