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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 831 | . . . 4 DECID | |
3 | 1, 2 | ax-mp 5 | . . 3 |
4 | vprc 4121 | . . . . . . 7 | |
5 | df-v 2732 | . . . . . . . . 9 | |
6 | equid 1694 | . . . . . . . . . . 11 | |
7 | pm5.1im 172 | . . . . . . . . . . 11 | |
8 | 6, 7 | ax-mp 5 | . . . . . . . . . 10 |
9 | 8 | abbidv 2288 | . . . . . . . . 9 |
10 | 5, 9 | eqtr2id 2216 | . . . . . . . 8 |
11 | 10 | eleq1d 2239 | . . . . . . 7 |
12 | 4, 11 | mtbiri 670 | . . . . . 6 |
13 | 12 | con2i 622 | . . . . 5 |
14 | vex 2733 | . . . . . . . . . 10 | |
15 | biidd 171 | . . . . . . . . . 10 | |
16 | 14, 15 | elab 2874 | . . . . . . . . 9 |
17 | 16 | notbii 663 | . . . . . . . 8 |
18 | 17 | biimpri 132 | . . . . . . 7 |
19 | 18 | eq0rdv 3459 | . . . . . 6 |
20 | 0ex 4116 | . . . . . 6 | |
21 | 19, 20 | eqeltrdi 2261 | . . . . 5 |
22 | 13, 21 | impbii 125 | . . . 4 |
23 | 22 | notbii 663 | . . . 4 |
24 | 22, 23 | orbi12i 759 | . . 3 |
25 | 3, 24 | mpbi 144 | . 2 |
26 | df-dc 830 | . 2 DECID | |
27 | 25, 26 | mpbir 145 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 wo 703 DECID wdc 829 wcel 2141 cab 2156 cvv 2730 c0 3414 |
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-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-in 3127 df-ss 3134 df-nul 3415 |
This theorem is referenced by: (None) |
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