New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > elsymdif | GIF version |
Description: Membership in symmetric difference. (Contributed by SF, 10-Jan-2015.) |
Ref | Expression |
---|---|
elsymdif | ⊢ (A ∈ (B ⊕ C) ↔ ¬ (A ∈ B ↔ A ∈ C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun 3220 | . . 3 ⊢ (A ∈ ((B ∖ C) ∪ (C ∖ B)) ↔ (A ∈ (B ∖ C) ∨ A ∈ (C ∖ B))) | |
2 | eldif 3221 | . . . 4 ⊢ (A ∈ (B ∖ C) ↔ (A ∈ B ∧ ¬ A ∈ C)) | |
3 | eldif 3221 | . . . 4 ⊢ (A ∈ (C ∖ B) ↔ (A ∈ C ∧ ¬ A ∈ B)) | |
4 | 2, 3 | orbi12i 507 | . . 3 ⊢ ((A ∈ (B ∖ C) ∨ A ∈ (C ∖ B)) ↔ ((A ∈ B ∧ ¬ A ∈ C) ∨ (A ∈ C ∧ ¬ A ∈ B))) |
5 | 1, 4 | bitri 240 | . 2 ⊢ (A ∈ ((B ∖ C) ∪ (C ∖ B)) ↔ ((A ∈ B ∧ ¬ A ∈ C) ∨ (A ∈ C ∧ ¬ A ∈ B))) |
6 | df-symdif 3216 | . . 3 ⊢ (B ⊕ C) = ((B ∖ C) ∪ (C ∖ B)) | |
7 | 6 | eleq2i 2417 | . 2 ⊢ (A ∈ (B ⊕ C) ↔ A ∈ ((B ∖ C) ∪ (C ∖ B))) |
8 | xor 861 | . 2 ⊢ (¬ (A ∈ B ↔ A ∈ C) ↔ ((A ∈ B ∧ ¬ A ∈ C) ∨ (A ∈ C ∧ ¬ A ∈ B))) | |
9 | 5, 7, 8 | 3bitr4i 268 | 1 ⊢ (A ∈ (B ⊕ C) ↔ ¬ (A ∈ B ↔ A ∈ C)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 176 ∨ wo 357 ∧ wa 358 ∈ wcel 1710 ∖ cdif 3206 ∪ cun 3207 ⊕ csymdif 3209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 |
This theorem is referenced by: opkelimagekg 4271 dfaddc2 4381 nnsucelrlem1 4424 ltfinex 4464 eqpwrelk 4478 eqpw1relk 4479 eqtfinrelk 4486 evenfinex 4503 oddfinex 4504 evenodddisjlem1 4515 nnadjoinlem1 4519 srelk 4524 tfinnnlem1 4533 dfop2lem1 4573 setconslem2 4732 setconslem3 4733 setconslem7 4737 dfswap2 4741 brimage 5793 releqel 5807 releqmpt2 5809 extex 5915 qsexg 5982 ovcelem1 6171 tcfnex 6244 |
Copyright terms: Public domain | W3C validator |