![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4101 | . 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) | |
2 | nfsymdif.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfsymdif.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
4 | 2, 3 | nfdif 3988 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵) |
5 | 3, 2 | nfdif 3988 | . . 3 ⊢ Ⅎ𝑥(𝐵 ∖ 𝐴) |
6 | 4, 5 | nfun 4026 | . 2 ⊢ Ⅎ𝑥((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) |
7 | 1, 6 | nfcxfr 2924 | 1 ⊢ Ⅎ𝑥(𝐴 △ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2910 ∖ cdif 3822 ∪ cun 3823 △ csymdif 4100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-rab 3091 df-dif 3828 df-un 3830 df-symdif 4101 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |