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Mirrors > Home > MPE Home > Th. List > symdifv | Structured version Visualization version GIF version |
Description: Symmetric difference with the universal class. (Contributed by Scott Fenton, 24-Apr-2012.) |
Ref | Expression |
---|---|
symdifv | ⊢ (𝐴 △ V) = (V ∖ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-symdif 3806 | . 2 ⊢ (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) | |
2 | ssv 3588 | . . . . 5 ⊢ 𝐴 ⊆ V | |
3 | ssdif0 3896 | . . . . 5 ⊢ (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅) | |
4 | 2, 3 | mpbi 219 | . . . 4 ⊢ (𝐴 ∖ V) = ∅ |
5 | 4 | uneq1i 3725 | . . 3 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴)) |
6 | uncom 3719 | . . . 4 ⊢ (∅ ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ ∅) | |
7 | un0 3919 | . . . 4 ⊢ ((V ∖ 𝐴) ∪ ∅) = (V ∖ 𝐴) | |
8 | 6, 7 | eqtri 2632 | . . 3 ⊢ (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴) |
9 | 5, 8 | eqtri 2632 | . 2 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (V ∖ 𝐴) |
10 | 1, 9 | eqtri 2632 | 1 ⊢ (𝐴 △ V) = (V ∖ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 △ csymdif 3805 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-symdif 3806 df-nul 3875 |
This theorem is referenced by: (None) |
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