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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 4222 | . 2 ⊢ (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) | |
2 | ssv 3994 | . . . . 5 ⊢ 𝐴 ⊆ V | |
3 | ssdif0 4326 | . . . . 5 ⊢ (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅) | |
4 | 2, 3 | mpbi 232 | . . . 4 ⊢ (𝐴 ∖ V) = ∅ |
5 | 4 | uneq1i 4138 | . . 3 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴)) |
6 | uncom 4132 | . . . 4 ⊢ (∅ ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ ∅) | |
7 | un0 4347 | . . . 4 ⊢ ((V ∖ 𝐴) ∪ ∅) = (V ∖ 𝐴) | |
8 | 6, 7 | eqtri 2847 | . . 3 ⊢ (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴) |
9 | 5, 8 | eqtri 2847 | . 2 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (V ∖ 𝐴) |
10 | 1, 9 | eqtri 2847 | 1 ⊢ (𝐴 △ V) = (V ∖ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 Vcvv 3497 ∖ cdif 3936 ∪ cun 3937 ⊆ wss 3939 △ csymdif 4221 ∅c0 4294 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-symdif 4222 df-nul 4295 |
This theorem is referenced by: (None) |
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