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Mirrors > Home > MPE Home > Th. List > symdifv | Structured version Visualization version GIF version |
Description: The symmetric difference with the universal class is the complement. (Contributed by Scott Fenton, 24-Apr-2012.) |
Ref | Expression |
---|---|
symdifv | ⊢ (𝐴 △ V) = (V ∖ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-symdif 4157 | . 2 ⊢ (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) | |
2 | ssv 3925 | . . . . 5 ⊢ 𝐴 ⊆ V | |
3 | ssdif0 4278 | . . . . 5 ⊢ (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅) | |
4 | 2, 3 | mpbi 233 | . . . 4 ⊢ (𝐴 ∖ V) = ∅ |
5 | 4 | uneq1i 4073 | . . 3 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴)) |
6 | uncom 4067 | . . . 4 ⊢ (∅ ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ ∅) | |
7 | un0 4305 | . . . 4 ⊢ ((V ∖ 𝐴) ∪ ∅) = (V ∖ 𝐴) | |
8 | 6, 7 | eqtri 2765 | . . 3 ⊢ (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴) |
9 | 5, 8 | eqtri 2765 | . 2 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (V ∖ 𝐴) |
10 | 1, 9 | eqtri 2765 | 1 ⊢ (𝐴 △ V) = (V ∖ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 Vcvv 3408 ∖ cdif 3863 ∪ cun 3864 ⊆ wss 3866 △ csymdif 4156 ∅c0 4237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-symdif 4157 df-nul 4238 |
This theorem is referenced by: (None) |
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