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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 4259 | . 2 ⊢ (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) | |
2 | ssv 4020 | . . . . 5 ⊢ 𝐴 ⊆ V | |
3 | ssdif0 4372 | . . . . 5 ⊢ (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅) | |
4 | 2, 3 | mpbi 230 | . . . 4 ⊢ (𝐴 ∖ V) = ∅ |
5 | 4 | uneq1i 4174 | . . 3 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴)) |
6 | uncom 4168 | . . . 4 ⊢ (∅ ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ ∅) | |
7 | un0 4400 | . . . 4 ⊢ ((V ∖ 𝐴) ∪ ∅) = (V ∖ 𝐴) | |
8 | 6, 7 | eqtri 2763 | . . 3 ⊢ (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴) |
9 | 5, 8 | eqtri 2763 | . 2 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (V ∖ 𝐴) |
10 | 1, 9 | eqtri 2763 | 1 ⊢ (𝐴 △ V) = (V ∖ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 ⊆ wss 3963 △ csymdif 4258 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-symdif 4259 df-nul 4340 |
This theorem is referenced by: (None) |
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