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Mirrors > Home > MPE Home > Th. List > symdifid | Structured version Visualization version GIF version |
Description: The symmetric difference of a class with itself is the empty class. (Contributed by Scott Fenton, 25-Apr-2012.) |
Ref | Expression |
---|---|
symdifid | ⊢ (𝐴 △ 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-symdif 4242 | . 2 ⊢ (𝐴 △ 𝐴) = ((𝐴 ∖ 𝐴) ∪ (𝐴 ∖ 𝐴)) | |
2 | difid 4370 | . . 3 ⊢ (𝐴 ∖ 𝐴) = ∅ | |
3 | 2, 2 | uneq12i 4161 | . 2 ⊢ ((𝐴 ∖ 𝐴) ∪ (𝐴 ∖ 𝐴)) = (∅ ∪ ∅) |
4 | un0 4390 | . 2 ⊢ (∅ ∪ ∅) = ∅ | |
5 | 1, 3, 4 | 3eqtri 2765 | 1 ⊢ (𝐴 △ 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∖ cdif 3945 ∪ cun 3946 △ csymdif 4241 ∅c0 4322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3951 df-un 3953 df-symdif 4242 df-nul 4323 |
This theorem is referenced by: (None) |
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