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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 4148 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∖ 𝐴) ∪ (𝐴 ∖ 𝐵)) | |
2 | df-symdif 4237 | . 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) | |
3 | df-symdif 4237 | . 2 ⊢ (𝐵 △ 𝐴) = ((𝐵 ∖ 𝐴) ∪ (𝐴 ∖ 𝐵)) | |
4 | 1, 2, 3 | 3eqtr4i 2764 | 1 ⊢ (𝐴 △ 𝐵) = (𝐵 △ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∖ cdif 3940 ∪ cun 3941 △ 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-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-un 3948 df-symdif 4237 |
This theorem is referenced by: symdifeq2 4240 |
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