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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 4253 | . 2 ⊢ (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) | |
| 2 | ssv 4008 | . . . . 5 ⊢ 𝐴 ⊆ V | |
| 3 | ssdif0 4366 | . . . . 5 ⊢ (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅) | |
| 4 | 2, 3 | mpbi 230 | . . . 4 ⊢ (𝐴 ∖ V) = ∅ |
| 5 | 4 | uneq1i 4164 | . . 3 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴)) |
| 6 | uncom 4158 | . . . 4 ⊢ (∅ ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ ∅) | |
| 7 | un0 4394 | . . . 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 1540 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 △ csymdif 4252 ∅c0 4333 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-symdif 4253 df-nul 4334 |
| This theorem is referenced by: (None) |
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