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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 3666 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 17-Jun-2022.) |
Ref | Expression |
---|---|
dcdifsnid | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difsnss 3666 | . . 3 | |
2 | 1 | adantl 275 | . 2 DECID |
3 | simpr 109 | . . . . . . 7 DECID | |
4 | velsn 3544 | . . . . . . 7 | |
5 | 3, 4 | sylibr 133 | . . . . . 6 DECID |
6 | elun2 3244 | . . . . . 6 | |
7 | 5, 6 | syl 14 | . . . . 5 DECID |
8 | simplr 519 | . . . . . . 7 DECID | |
9 | simpr 109 | . . . . . . . 8 DECID | |
10 | 9, 4 | sylnibr 666 | . . . . . . 7 DECID |
11 | 8, 10 | eldifd 3081 | . . . . . 6 DECID |
12 | elun1 3243 | . . . . . 6 | |
13 | 11, 12 | syl 14 | . . . . 5 DECID |
14 | simpll 518 | . . . . . . 7 DECID DECID | |
15 | simpr 109 | . . . . . . . 8 DECID | |
16 | simplr 519 | . . . . . . . 8 DECID | |
17 | equequ1 1688 | . . . . . . . . . 10 | |
18 | 17 | dcbid 823 | . . . . . . . . 9 DECID DECID |
19 | eqeq2 2149 | . . . . . . . . . 10 | |
20 | 19 | dcbid 823 | . . . . . . . . 9 DECID DECID |
21 | 18, 20 | rspc2v 2802 | . . . . . . . 8 DECID DECID |
22 | 15, 16, 21 | syl2anc 408 | . . . . . . 7 DECID DECID DECID |
23 | 14, 22 | mpd 13 | . . . . . 6 DECID DECID |
24 | exmiddc 821 | . . . . . 6 DECID | |
25 | 23, 24 | syl 14 | . . . . 5 DECID |
26 | 7, 13, 25 | mpjaodan 787 | . . . 4 DECID |
27 | 26 | ex 114 | . . 3 DECID |
28 | 27 | ssrdv 3103 | . 2 DECID |
29 | 2, 28 | eqssd 3114 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 DECID wdc 819 wceq 1331 wcel 1480 wral 2416 cdif 3068 cun 3069 wss 3071 csn 3527 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 |
This theorem is referenced by: fnsnsplitdc 6401 nndifsnid 6403 fidifsnid 6765 undifdc 6812 |
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