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Mirrors > Home > MPE Home > Th. List > symdifcom | Structured version Visualization version GIF version |
Description: Symmetric difference commutes. (Contributed by Scott Fenton, 24-Apr-2012.) |
Ref | Expression |
---|---|
symdifcom | ⊢ (𝐴 △ 𝐵) = (𝐵 △ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 3757 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∖ 𝐴) ∪ (𝐴 ∖ 𝐵)) | |
2 | df-symdif 3844 | . 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) | |
3 | df-symdif 3844 | . 2 ⊢ (𝐵 △ 𝐴) = ((𝐵 ∖ 𝐴) ∪ (𝐴 ∖ 𝐵)) | |
4 | 1, 2, 3 | 3eqtr4i 2654 | 1 ⊢ (𝐴 △ 𝐵) = (𝐵 △ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∖ cdif 3571 ∪ cun 3572 △ csymdif 3843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-un 3579 df-symdif 3844 |
This theorem is referenced by: symdifeq2 3847 |
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