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Mirrors > Home > ILE Home > Th. List > exmidel | Unicode version |
Description: Excluded middle is equivalent to decidability of membership for two arbitrary sets. (Contributed by Jim Kingdon, 18-Jun-2022.) |
Ref | Expression |
---|---|
exmidel | EXMID DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmidexmid 4120 | . . 3 EXMID DECID | |
2 | 1 | alrimivv 1847 | . 2 EXMID DECID |
3 | 0ex 4055 | . . . 4 | |
4 | eleq1 2202 | . . . . . 6 | |
5 | 4 | dcbid 823 | . . . . 5 DECID DECID |
6 | 5 | albidv 1796 | . . . 4 DECID DECID |
7 | 3, 6 | spcv 2779 | . . 3 DECID DECID |
8 | exmid0el 4127 | . . 3 EXMID DECID | |
9 | 7, 8 | sylibr 133 | . 2 DECID EXMID |
10 | 2, 9 | impbii 125 | 1 EXMID DECID |
Colors of variables: wff set class |
Syntax hints: wb 104 DECID wdc 819 wal 1329 wceq 1331 wcel 1480 c0 3363 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-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: (None) |
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