| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > symdif0 | Structured version Visualization version GIF version | ||
| Description: Symmetric difference with the empty class. The empty class is the identity element for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.) |
| Ref | Expression |
|---|---|
| symdif0 | ⊢ (𝐴 △ ∅) = 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-symdif 4253 | . 2 ⊢ (𝐴 △ ∅) = ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴)) | |
| 2 | dif0 4378 | . . 3 ⊢ (𝐴 ∖ ∅) = 𝐴 | |
| 3 | 0dif 4405 | . . 3 ⊢ (∅ ∖ 𝐴) = ∅ | |
| 4 | 2, 3 | uneq12i 4166 | . 2 ⊢ ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴)) = (𝐴 ∪ ∅) |
| 5 | un0 4394 | . 2 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
| 6 | 1, 4, 5 | 3eqtri 2769 | 1 ⊢ (𝐴 △ ∅) = 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∖ cdif 3948 ∪ cun 3949 △ csymdif 4252 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-symdif 4253 df-nul 4334 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |