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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 4014 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∖ 𝐴) ∪ (𝐴 ∖ 𝐵)) | |
2 | df-symdif 4101 | . 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) | |
3 | df-symdif 4101 | . 2 ⊢ (𝐵 △ 𝐴) = ((𝐵 ∖ 𝐴) ∪ (𝐴 ∖ 𝐵)) | |
4 | 1, 2, 3 | 3eqtr4i 2806 | 1 ⊢ (𝐴 △ 𝐵) = (𝐵 △ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∖ cdif 3822 ∪ cun 3823 △ csymdif 4100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-v 3411 df-un 3830 df-symdif 4101 |
This theorem is referenced by: symdifeq2 4104 |
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