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Mirrors > Home > NFE Home > Th. List > symdifex | GIF version |
Description: The symmetric difference of two sets is a set. (Contributed by SF, 12-Jan-2015.) |
Ref | Expression |
---|---|
boolex.1 | ⊢ A ∈ V |
boolex.2 | ⊢ B ∈ V |
Ref | Expression |
---|---|
symdifex | ⊢ (A ⊕ B) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | boolex.1 | . 2 ⊢ A ∈ V | |
2 | boolex.2 | . 2 ⊢ B ∈ V | |
3 | symdifexg 4103 | . 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A ⊕ B) ∈ V) | |
4 | 1, 2, 3 | mp2an 653 | 1 ⊢ (A ⊕ B) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1710 Vcvv 2859 ⊕ csymdif 3209 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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 ax-nin 4078 |
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: addcexlem 4382 nnsucelrlem1 4424 ltfinex 4464 ncfinraiselem2 4480 ncfinlowerlem1 4482 tfinrelkex 4487 evenfinex 4503 oddfinex 4504 evenodddisjlem1 4515 nnadjoinlem1 4519 nnpweqlem1 4522 srelkex 4525 tfinnnlem1 4533 opexg 4587 proj2exg 4592 setconslem5 4735 1stex 4739 swapex 4742 mptexlem 5810 mpt2exlem 5811 extex 5915 ovcelem1 6171 ceex 6174 tcfnex 6244 nmembers1lem1 6268 nchoicelem11 6299 nchoicelem16 6304 nchoicelem18 6306 |
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