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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 807 | . . . . . . . 8 DECID | |
4 | 2, 3 | ax-mp 5 | . . . . . . 7 |
5 | 4 | sbcth 2895 | . . . . . 6 |
6 | 1, 5 | ax-mp 5 | . . . . 5 |
7 | sbcor 2925 | . . . . 5 | |
8 | 6, 7 | mpbi 144 | . . . 4 |
9 | ralsns 3532 | . . . . . 6 | |
10 | 1, 9 | ax-mp 5 | . . . . 5 |
11 | ralsns 3532 | . . . . . 6 | |
12 | 1, 11 | ax-mp 5 | . . . . 5 |
13 | 10, 12 | orbi12i 738 | . . . 4 |
14 | 8, 13 | mpbir 145 | . . 3 |
15 | rabeq0 3362 | . . . 4 | |
16 | eqcom 2119 | . . . . 5 | |
17 | rabid2 2584 | . . . . 5 | |
18 | 16, 17 | bitri 183 | . . . 4 |
19 | 15, 18 | orbi12i 738 | . . 3 |
20 | 14, 19 | mpbir 145 | . 2 |
21 | eqeq1 2124 | . . 3 | |
22 | eqeq1 2124 | . . 3 | |
23 | 21, 22 | orbi12d 767 | . 2 |
24 | 20, 23 | mpbiri 167 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 wo 682 DECID wdc 804 wceq 1316 wcel 1465 wral 2393 crab 2397 cvv 2660 wsbc 2882 c0 3333 csn 3497 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-nul 3334 df-sn 3503 |
This theorem is referenced by: (None) |
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