Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > exmidn0m | Unicode version |
Description: Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.) |
Ref | Expression |
---|---|
exmidn0m | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . 5 EXMID | |
2 | 1 | olcd 729 | . . . 4 EXMID |
3 | notm0 3434 | . . . . . . 7 | |
4 | 3 | biimpi 119 | . . . . . 6 |
5 | 4 | adantl 275 | . . . . 5 EXMID |
6 | 5 | orcd 728 | . . . 4 EXMID |
7 | exmidexmid 4180 | . . . . 5 EXMID DECID | |
8 | exmiddc 831 | . . . . 5 DECID | |
9 | 7, 8 | syl 14 | . . . 4 EXMID |
10 | 2, 6, 9 | mpjaodan 793 | . . 3 EXMID |
11 | 10 | alrimiv 1867 | . 2 EXMID |
12 | orc 707 | . . . . . 6 | |
13 | 12 | a1d 22 | . . . . 5 |
14 | sssnm 3739 | . . . . . . . 8 | |
15 | 14 | biimpa 294 | . . . . . . 7 |
16 | 15 | olcd 729 | . . . . . 6 |
17 | 16 | ex 114 | . . . . 5 |
18 | 13, 17 | jaoi 711 | . . . 4 |
19 | 18 | alimi 1448 | . . 3 |
20 | exmid01 4182 | . . 3 EXMID | |
21 | 19, 20 | sylibr 133 | . 2 EXMID |
22 | 11, 21 | impbii 125 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 703 DECID wdc 829 wal 1346 wceq 1348 wex 1485 wss 3121 c0 3414 csn 3581 EXMIDwem 4178 |
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-14 2144 ax-ext 2152 ax-sep 4105 ax-nul 4113 ax-pow 4158 |
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-rab 2457 df-v 2732 df-dif 3123 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-exmid 4179 |
This theorem is referenced by: exmidsssn 4186 |
Copyright terms: Public domain | W3C validator |