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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 4180 | . . 3 EXMID DECID | |
2 | 1 | alrimivv 1868 | . 2 EXMID DECID |
3 | 0ex 4114 | . . . 4 | |
4 | eleq1 2233 | . . . . . 6 | |
5 | 4 | dcbid 833 | . . . . 5 DECID DECID |
6 | 5 | albidv 1817 | . . . 4 DECID DECID |
7 | 3, 6 | spcv 2824 | . . 3 DECID DECID |
8 | exmid0el 4188 | . . 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 829 wal 1346 wceq 1348 wcel 2141 c0 3414 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-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: (None) |
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