Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4221 | . 2 ⊢ (𝐴 △ 𝐴) = ((𝐴 ∖ 𝐴) ∪ (𝐴 ∖ 𝐴)) | |
2 | difid 4332 | . . 3 ⊢ (𝐴 ∖ 𝐴) = ∅ | |
3 | 2, 2 | uneq12i 4139 | . 2 ⊢ ((𝐴 ∖ 𝐴) ∪ (𝐴 ∖ 𝐴)) = (∅ ∪ ∅) |
4 | un0 4346 | . 2 ⊢ (∅ ∪ ∅) = ∅ | |
5 | 1, 3, 4 | 3eqtri 2850 | 1 ⊢ (𝐴 △ 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3935 ∪ cun 3936 △ csymdif 4220 ∅c0 4293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-symdif 4221 df-nul 4294 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |