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Mirrors > Home > MPE Home > Th. List > nfsymdif | Structured version Visualization version GIF version |
Description: Hypothesis builder for symmetric difference. (Contributed by Scott Fenton, 19-Feb-2013.) (Revised by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
nfsymdif.1 | ⊢ Ⅎ𝑥𝐴 |
nfsymdif.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfsymdif | ⊢ Ⅎ𝑥(𝐴 △ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-symdif 3806 | . 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) | |
2 | nfsymdif.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfsymdif.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
4 | 2, 3 | nfdif 3693 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵) |
5 | 3, 2 | nfdif 3693 | . . 3 ⊢ Ⅎ𝑥(𝐵 ∖ 𝐴) |
6 | 4, 5 | nfun 3731 | . 2 ⊢ Ⅎ𝑥((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) |
7 | 1, 6 | nfcxfr 2749 | 1 ⊢ Ⅎ𝑥(𝐴 △ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2738 ∖ cdif 3537 ∪ cun 3538 △ csymdif 3805 |
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-rab 2905 df-dif 3543 df-un 3545 df-symdif 3806 |
This theorem is referenced by: (None) |
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