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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 4248 | . 2 ⊢ (𝑥 ∈ (𝐴 △ 𝐵) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)) | |
2 | 1 | eqabi 2862 | 1 ⊢ (𝐴 △ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)} |
Colors of variables: wff setvar class |
Syntax hints: ⊻ wxo 1505 = wceq 1534 ∈ wcel 2099 {cab 2703 △ csymdif 4240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-xor 1506 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-v 3464 df-dif 3949 df-un 3951 df-symdif 4241 |
This theorem is referenced by: (None) |
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