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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 3632 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 3632 |
. . 3
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2 | 1 | adantl 273 |
. 2
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3 | simpr 109 |
. . . . . . 7
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4 | velsn 3510 |
. . . . . . 7
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5 | 3, 4 | sylibr 133 |
. . . . . 6
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6 | elun2 3210 |
. . . . . 6
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7 | 5, 6 | syl 14 |
. . . . 5
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8 | simplr 502 |
. . . . . . 7
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9 | simpr 109 |
. . . . . . . 8
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10 | 9, 4 | sylnibr 649 |
. . . . . . 7
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11 | 8, 10 | eldifd 3047 |
. . . . . 6
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12 | elun1 3209 |
. . . . . 6
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13 | 11, 12 | syl 14 |
. . . . 5
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14 | simpll 501 |
. . . . . . 7
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15 | simpr 109 |
. . . . . . . 8
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16 | simplr 502 |
. . . . . . . 8
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17 | equequ1 1671 |
. . . . . . . . . 10
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18 | 17 | dcbid 806 |
. . . . . . . . 9
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19 | eqeq2 2124 |
. . . . . . . . . 10
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20 | 19 | dcbid 806 |
. . . . . . . . 9
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21 | 18, 20 | rspc2v 2772 |
. . . . . . . 8
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22 | 15, 16, 21 | syl2anc 406 |
. . . . . . 7
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23 | 14, 22 | mpd 13 |
. . . . . 6
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24 | exmiddc 804 |
. . . . . 6
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25 | 23, 24 | syl 14 |
. . . . 5
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26 | 7, 13, 25 | mpjaodan 770 |
. . . 4
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27 | 26 | ex 114 |
. . 3
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28 | 27 | ssrdv 3069 |
. 2
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29 | 2, 28 | eqssd 3080 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-sn 3499 |
This theorem is referenced by: fnsnsplitdc 6355 nndifsnid 6357 fidifsnid 6718 undifdc 6765 |
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