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Mirrors > Home > MPE Home > Th. List > symdif0 | Structured version Visualization version GIF version |
Description: Symmetric difference with the empty class. (Contributed by Scott Fenton, 24-Apr-2012.) |
Ref | Expression |
---|---|
symdif0 | ⊢ (𝐴 △ ∅) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-symdif 3844 | . 2 ⊢ (𝐴 △ ∅) = ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴)) | |
2 | dif0 3950 | . . 3 ⊢ (𝐴 ∖ ∅) = 𝐴 | |
3 | 0dif 3977 | . . 3 ⊢ (∅ ∖ 𝐴) = ∅ | |
4 | 2, 3 | uneq12i 3765 | . 2 ⊢ ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴)) = (𝐴 ∪ ∅) |
5 | un0 3967 | . 2 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
6 | 1, 4, 5 | 3eqtri 2648 | 1 ⊢ (𝐴 △ ∅) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∖ cdif 3571 ∪ cun 3572 △ csymdif 3843 ∅c0 3915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-symdif 3844 df-nul 3916 |
This theorem is referenced by: (None) |
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