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Mirrors > Home > NFE Home > Th. List > symdifexg | GIF version |
Description: The symmetric difference of two sets is a set. (Contributed by SF, 12-Jan-2015.) |
Ref | Expression |
---|---|
symdifexg | ⊢ ((A ∈ V ∧ B ∈ W) → (A ⊕ B) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-symdif 3216 | . 2 ⊢ (A ⊕ B) = ((A ∖ B) ∪ (B ∖ A)) | |
2 | difexg 4102 | . . 3 ⊢ ((A ∈ V ∧ B ∈ W) → (A ∖ B) ∈ V) | |
3 | difexg 4102 | . . . 4 ⊢ ((B ∈ W ∧ A ∈ V) → (B ∖ A) ∈ V) | |
4 | 3 | ancoms 439 | . . 3 ⊢ ((A ∈ V ∧ B ∈ W) → (B ∖ A) ∈ V) |
5 | unexg 4101 | . . 3 ⊢ (((A ∖ B) ∈ V ∧ (B ∖ A) ∈ V) → ((A ∖ B) ∪ (B ∖ A)) ∈ V) | |
6 | 2, 4, 5 | syl2anc 642 | . 2 ⊢ ((A ∈ V ∧ B ∈ W) → ((A ∖ B) ∪ (B ∖ A)) ∈ V) |
7 | 1, 6 | syl5eqel 2437 | 1 ⊢ ((A ∈ V ∧ B ∈ W) → (A ⊕ B) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∈ wcel 1710 Vcvv 2859 ∖ cdif 3206 ∪ cun 3207 ⊕ 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: symdifex 4108 imagekexg 4311 imageexg 5800 qsexg 5982 |
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