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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 724 | . . . 4 EXMID |
3 | notm0 3388 | . . . . . . 7 | |
4 | 3 | biimpi 119 | . . . . . 6 |
5 | 4 | adantl 275 | . . . . 5 EXMID |
6 | 5 | orcd 723 | . . . 4 EXMID |
7 | exmidexmid 4128 | . . . . 5 EXMID DECID | |
8 | exmiddc 822 | . . . . 5 DECID | |
9 | 7, 8 | syl 14 | . . . 4 EXMID |
10 | 2, 6, 9 | mpjaodan 788 | . . 3 EXMID |
11 | 10 | alrimiv 1847 | . 2 EXMID |
12 | orc 702 | . . . . . 6 | |
13 | 12 | a1d 22 | . . . . 5 |
14 | sssnm 3689 | . . . . . . . 8 | |
15 | 14 | biimpa 294 | . . . . . . 7 |
16 | 15 | olcd 724 | . . . . . 6 |
17 | 16 | ex 114 | . . . . 5 |
18 | 13, 17 | jaoi 706 | . . . 4 |
19 | 18 | alimi 1432 | . . 3 |
20 | exmid01 4129 | . . 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 698 DECID wdc 820 wal 1330 wceq 1332 wex 1469 wss 3076 c0 3368 csn 3532 EXMIDwem 4126 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rab 2426 df-v 2691 df-dif 3078 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-exmid 4127 |
This theorem is referenced by: exmidsssn 4133 |
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