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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 |
. . . . 5
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2 | 1 | olcd 694 |
. . . 4
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3 | notm0 3330 |
. . . . . . 7
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4 | 3 | biimpi 119 |
. . . . . 6
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5 | 4 | adantl 273 |
. . . . 5
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6 | 5 | orcd 693 |
. . . 4
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7 | exmidexmid 4060 |
. . . . 5
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8 | exmiddc 788 |
. . . . 5
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9 | 7, 8 | syl 14 |
. . . 4
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10 | 2, 6, 9 | mpjaodan 753 |
. . 3
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11 | 10 | alrimiv 1813 |
. 2
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12 | orc 674 |
. . . . . 6
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13 | 12 | a1d 22 |
. . . . 5
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14 | sssnm 3628 |
. . . . . . . 8
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15 | 14 | biimpa 292 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 15 | olcd 694 |
. . . . . 6
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17 | 16 | ex 114 |
. . . . 5
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18 | 13, 17 | jaoi 677 |
. . . 4
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19 | 18 | alimi 1399 |
. . 3
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20 | exmid01 4061 |
. . 3
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21 | 19, 20 | sylibr 133 |
. 2
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22 | 11, 21 | impbii 125 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-nul 3994 ax-pow 4038 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-rab 2384 df-v 2643 df-dif 3023 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-exmid 4059 |
This theorem is referenced by: exmidsssn 4063 |
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