Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > dcextest | Unicode version |
Description: If it is decidable whether is a set, then is decidable (where does not occur in ). From this fact, we can deduce (outside the formal system, since we cannot quantify over classes) that if it is decidable whether any class is a set, then "weak excluded middle" (that is, any negated proposition is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.) |
Ref | Expression |
---|---|
dcextest.ex | DECID |
Ref | Expression |
---|---|
dcextest | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dcextest.ex | . . . 4 DECID | |
2 | exmiddc 822 | . . . 4 DECID | |
3 | 1, 2 | ax-mp 5 | . . 3 |
4 | vprc 4097 | . . . . . . 7 | |
5 | df-v 2714 | . . . . . . . . 9 | |
6 | equid 1681 | . . . . . . . . . . 11 | |
7 | pm5.1im 172 | . . . . . . . . . . 11 | |
8 | 6, 7 | ax-mp 5 | . . . . . . . . . 10 |
9 | 8 | abbidv 2275 | . . . . . . . . 9 |
10 | 5, 9 | eqtr2id 2203 | . . . . . . . 8 |
11 | 10 | eleq1d 2226 | . . . . . . 7 |
12 | 4, 11 | mtbiri 665 | . . . . . 6 |
13 | 12 | con2i 617 | . . . . 5 |
14 | vex 2715 | . . . . . . . . . 10 | |
15 | biidd 171 | . . . . . . . . . 10 | |
16 | 14, 15 | elab 2856 | . . . . . . . . 9 |
17 | 16 | notbii 658 | . . . . . . . 8 |
18 | 17 | biimpri 132 | . . . . . . 7 |
19 | 18 | eq0rdv 3438 | . . . . . 6 |
20 | 0ex 4092 | . . . . . 6 | |
21 | 19, 20 | eqeltrdi 2248 | . . . . 5 |
22 | 13, 21 | impbii 125 | . . . 4 |
23 | 22 | notbii 658 | . . . 4 |
24 | 22, 23 | orbi12i 754 | . . 3 |
25 | 3, 24 | mpbi 144 | . 2 |
26 | df-dc 821 | . 2 DECID | |
27 | 25, 26 | mpbir 145 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 wo 698 DECID wdc 820 wcel 2128 cab 2143 cvv 2712 c0 3394 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-dif 3104 df-in 3108 df-ss 3115 df-nul 3395 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |