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Mirrors > Home > ILE Home > Th. List > rabrsndc | Unicode version |
Description: A class abstraction over a decidable proposition restricted to a singleton is either the empty set or the singleton itself. (Contributed by Jim Kingdon, 8-Aug-2018.) |
Ref | Expression |
---|---|
rabrsndc.1 | |
rabrsndc.2 | DECID |
Ref | Expression |
---|---|
rabrsndc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabrsndc.1 | . . . . . 6 | |
2 | rabrsndc.2 | . . . . . . . 8 DECID | |
3 | pm2.1dc 745 | . . . . . . . 8 DECID | |
4 | 2, 3 | ax-mp 7 | . . . . . . 7 |
5 | 4 | sbcth 2777 | . . . . . 6 |
6 | 1, 5 | ax-mp 7 | . . . . 5 |
7 | sbcor 2807 | . . . . 5 | |
8 | 6, 7 | mpbi 133 | . . . 4 |
9 | ralsns 3408 | . . . . . 6 | |
10 | 1, 9 | ax-mp 7 | . . . . 5 |
11 | ralsns 3408 | . . . . . 6 | |
12 | 1, 11 | ax-mp 7 | . . . . 5 |
13 | 10, 12 | orbi12i 681 | . . . 4 |
14 | 8, 13 | mpbir 134 | . . 3 |
15 | rabeq0 3247 | . . . 4 | |
16 | eqcom 2042 | . . . . 5 | |
17 | rabid2 2486 | . . . . 5 | |
18 | 16, 17 | bitri 173 | . . . 4 |
19 | 15, 18 | orbi12i 681 | . . 3 |
20 | 14, 19 | mpbir 134 | . 2 |
21 | eqeq1 2046 | . . 3 | |
22 | eqeq1 2046 | . . 3 | |
23 | 21, 22 | orbi12d 707 | . 2 |
24 | 20, 23 | mpbiri 157 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 98 wo 629 DECID wdc 742 wceq 1243 wcel 1393 wral 2306 crab 2310 cvv 2557 wsbc 2764 c0 3224 csn 3375 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-nul 3225 df-sn 3381 |
This theorem is referenced by: (None) |
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