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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 723 | . . . 4 EXMID |
3 | notm0 3383 | . . . . . . 7 | |
4 | 3 | biimpi 119 | . . . . . 6 |
5 | 4 | adantl 275 | . . . . 5 EXMID |
6 | 5 | orcd 722 | . . . 4 EXMID |
7 | exmidexmid 4120 | . . . . 5 EXMID DECID | |
8 | exmiddc 821 | . . . . 5 DECID | |
9 | 7, 8 | syl 14 | . . . 4 EXMID |
10 | 2, 6, 9 | mpjaodan 787 | . . 3 EXMID |
11 | 10 | alrimiv 1846 | . 2 EXMID |
12 | orc 701 | . . . . . 6 | |
13 | 12 | a1d 22 | . . . . 5 |
14 | sssnm 3681 | . . . . . . . 8 | |
15 | 14 | biimpa 294 | . . . . . . 7 |
16 | 15 | olcd 723 | . . . . . 6 |
17 | 16 | ex 114 | . . . . 5 |
18 | 13, 17 | jaoi 705 | . . . 4 |
19 | 18 | alimi 1431 | . . 3 |
20 | exmid01 4121 | . . 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 697 DECID wdc 819 wal 1329 wceq 1331 wex 1468 wss 3071 c0 3363 csn 3527 EXMIDwem 4118 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 |
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-rab 2425 df-v 2688 df-dif 3073 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-exmid 4119 |
This theorem is referenced by: exmidsssn 4125 |
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