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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 3724 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 3724 | . . 3 | |
2 | 1 | adantl 275 | . 2 DECID |
3 | simpr 109 | . . . . . . 7 DECID | |
4 | velsn 3598 | . . . . . . 7 | |
5 | 3, 4 | sylibr 133 | . . . . . 6 DECID |
6 | elun2 3295 | . . . . . 6 | |
7 | 5, 6 | syl 14 | . . . . 5 DECID |
8 | simplr 525 | . . . . . . 7 DECID | |
9 | simpr 109 | . . . . . . . 8 DECID | |
10 | 9, 4 | sylnibr 672 | . . . . . . 7 DECID |
11 | 8, 10 | eldifd 3131 | . . . . . 6 DECID |
12 | elun1 3294 | . . . . . 6 | |
13 | 11, 12 | syl 14 | . . . . 5 DECID |
14 | simpll 524 | . . . . . . 7 DECID DECID | |
15 | simpr 109 | . . . . . . . 8 DECID | |
16 | simplr 525 | . . . . . . . 8 DECID | |
17 | equequ1 1705 | . . . . . . . . . 10 | |
18 | 17 | dcbid 833 | . . . . . . . . 9 DECID DECID |
19 | eqeq2 2180 | . . . . . . . . . 10 | |
20 | 19 | dcbid 833 | . . . . . . . . 9 DECID DECID |
21 | 18, 20 | rspc2v 2847 | . . . . . . . 8 DECID DECID |
22 | 15, 16, 21 | syl2anc 409 | . . . . . . 7 DECID DECID DECID |
23 | 14, 22 | mpd 13 | . . . . . 6 DECID DECID |
24 | exmiddc 831 | . . . . . 6 DECID | |
25 | 23, 24 | syl 14 | . . . . 5 DECID |
26 | 7, 13, 25 | mpjaodan 793 | . . . 4 DECID |
27 | 26 | ex 114 | . . 3 DECID |
28 | 27 | ssrdv 3153 | . 2 DECID |
29 | 2, 28 | eqssd 3164 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 703 DECID wdc 829 wceq 1348 wcel 2141 wral 2448 cdif 3118 cun 3119 wss 3121 csn 3581 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 |
This theorem is referenced by: fnsnsplitdc 6482 nndifsnid 6484 fidifsnid 6847 undifdc 6899 |
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