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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 821 | . . . 4 DECID | |
3 | 1, 2 | ax-mp 5 | . . 3 |
4 | vprc 4060 | . . . . . . 7 | |
5 | df-v 2688 | . . . . . . . . 9 | |
6 | equid 1677 | . . . . . . . . . . 11 | |
7 | pm5.1im 172 | . . . . . . . . . . 11 | |
8 | 6, 7 | ax-mp 5 | . . . . . . . . . 10 |
9 | 8 | abbidv 2257 | . . . . . . . . 9 |
10 | 5, 9 | syl5req 2185 | . . . . . . . 8 |
11 | 10 | eleq1d 2208 | . . . . . . 7 |
12 | 4, 11 | mtbiri 664 | . . . . . 6 |
13 | 12 | con2i 616 | . . . . 5 |
14 | vex 2689 | . . . . . . . . . 10 | |
15 | biidd 171 | . . . . . . . . . 10 | |
16 | 14, 15 | elab 2828 | . . . . . . . . 9 |
17 | 16 | notbii 657 | . . . . . . . 8 |
18 | 17 | biimpri 132 | . . . . . . 7 |
19 | 18 | eq0rdv 3407 | . . . . . 6 |
20 | 0ex 4055 | . . . . . 6 | |
21 | 19, 20 | eqeltrdi 2230 | . . . . 5 |
22 | 13, 21 | impbii 125 | . . . 4 |
23 | 22 | notbii 657 | . . . 4 |
24 | 22, 23 | orbi12i 753 | . . 3 |
25 | 3, 24 | mpbi 144 | . 2 |
26 | df-dc 820 | . 2 DECID | |
27 | 25, 26 | mpbir 145 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 wo 697 DECID wdc 819 wcel 1480 cab 2125 cvv 2686 c0 3363 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-dif 3073 df-in 3077 df-ss 3084 df-nul 3364 |
This theorem is referenced by: (None) |
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