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Mirrors > Home > MPE Home > Th. List > dfsymdif3 | Structured version Visualization version GIF version |
Description: Alternate definition of the symmetric difference, given in Example 4.1 of [Stoll] p. 262 (the original definition corresponds to [Stoll] p. 13). (Contributed by NM, 17-Aug-2004.) (Revised by BJ, 30-Apr-2020.) |
Ref | Expression |
---|---|
dfsymdif3 | ⊢ (𝐴 △ 𝐵) = ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difin 4219 | . . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
2 | incom 4159 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
3 | 2 | difeq2i 4077 | . . . 4 ⊢ (𝐵 ∖ (𝐴 ∩ 𝐵)) = (𝐵 ∖ (𝐵 ∩ 𝐴)) |
4 | difin 4219 | . . . 4 ⊢ (𝐵 ∖ (𝐵 ∩ 𝐴)) = (𝐵 ∖ 𝐴) | |
5 | 3, 4 | eqtri 2764 | . . 3 ⊢ (𝐵 ∖ (𝐴 ∩ 𝐵)) = (𝐵 ∖ 𝐴) |
6 | 1, 5 | uneq12i 4119 | . 2 ⊢ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) |
7 | difundir 4238 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵)) = ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))) | |
8 | df-symdif 4200 | . 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) | |
9 | 6, 7, 8 | 3eqtr4ri 2775 | 1 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∖ cdif 3905 ∪ cun 3906 ∩ cin 3907 △ csymdif 4199 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-symdif 4200 |
This theorem is referenced by: (None) |
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