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Mirrors > Home > ILE Home > Th. List > dcdifsnid | Unicode version |
Description: If we remove a single element from a set with decidable equality then put it back in, we end up with the original set. This strengthens difsnss 3674 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 17-Jun-2022.) |
Ref | Expression |
---|---|
dcdifsnid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difsnss 3674 |
. . 3
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2 | 1 | adantl 275 |
. 2
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3 | simpr 109 |
. . . . . . 7
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4 | velsn 3549 |
. . . . . . 7
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5 | 3, 4 | sylibr 133 |
. . . . . 6
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6 | elun2 3249 |
. . . . . 6
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7 | 5, 6 | syl 14 |
. . . . 5
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8 | simplr 520 |
. . . . . . 7
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9 | simpr 109 |
. . . . . . . 8
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10 | 9, 4 | sylnibr 667 |
. . . . . . 7
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11 | 8, 10 | eldifd 3086 |
. . . . . 6
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12 | elun1 3248 |
. . . . . 6
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13 | 11, 12 | syl 14 |
. . . . 5
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14 | simpll 519 |
. . . . . . 7
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15 | simpr 109 |
. . . . . . . 8
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16 | simplr 520 |
. . . . . . . 8
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17 | equequ1 1689 |
. . . . . . . . . 10
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18 | 17 | dcbid 824 |
. . . . . . . . 9
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19 | eqeq2 2150 |
. . . . . . . . . 10
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20 | 19 | dcbid 824 |
. . . . . . . . 9
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21 | 18, 20 | rspc2v 2806 |
. . . . . . . 8
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22 | 15, 16, 21 | syl2anc 409 |
. . . . . . 7
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23 | 14, 22 | mpd 13 |
. . . . . 6
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24 | exmiddc 822 |
. . . . . 6
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25 | 23, 24 | syl 14 |
. . . . 5
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26 | 7, 13, 25 | mpjaodan 788 |
. . . 4
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27 | 26 | ex 114 |
. . 3
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28 | 27 | ssrdv 3108 |
. 2
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29 | 2, 28 | eqssd 3119 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 |
This theorem is referenced by: fnsnsplitdc 6409 nndifsnid 6411 fidifsnid 6773 undifdc 6820 |
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