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Mirrors > Home > ILE Home > Th. List > exmidexmid | Unicode version |
Description: EXMID implies that an
arbitrary proposition is decidable. That is,
EXMID captures the usual meaning of excluded middle when stated in terms
of propositions.
To get other propositional statements which are equivalent to excluded middle, combine this with notnotrdc 829, peircedc 900, or condc 839. (Contributed by Jim Kingdon, 18-Jun-2022.) |
Ref | Expression |
---|---|
exmidexmid | EXMID DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3213 | . . 3 | |
2 | df-exmid 4156 | . . . 4 EXMID DECID | |
3 | p0ex 4149 | . . . . . 6 | |
4 | 3 | rabex 4108 | . . . . 5 |
5 | sseq1 3151 | . . . . . 6 | |
6 | eleq2 2221 | . . . . . . 7 | |
7 | 6 | dcbid 824 | . . . . . 6 DECID DECID |
8 | 5, 7 | imbi12d 233 | . . . . 5 DECID DECID |
9 | 4, 8 | spcv 2806 | . . . 4 DECID DECID |
10 | 2, 9 | sylbi 120 | . . 3 EXMID DECID |
11 | 1, 10 | mpi 15 | . 2 EXMID DECID |
12 | 0ex 4091 | . . . . 5 | |
13 | 12 | snid 3591 | . . . 4 |
14 | biidd 171 | . . . . 5 | |
15 | 14 | elrab 2868 | . . . 4 |
16 | 13, 15 | mpbiran 925 | . . 3 |
17 | 16 | dcbii 826 | . 2 DECID DECID |
18 | 11, 17 | sylib 121 | 1 EXMID DECID |
Colors of variables: wff set class |
Syntax hints: wi 4 DECID wdc 820 wal 1333 wceq 1335 wcel 2128 crab 2439 wss 3102 c0 3394 csn 3560 EXMIDwem 4155 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rab 2444 df-v 2714 df-dif 3104 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-exmid 4156 |
This theorem is referenced by: exmidn0m 4162 exmid0el 4165 exmidel 4166 exmidundif 4167 exmidundifim 4168 sbthlemi3 6903 sbthlemi5 6905 sbthlemi6 6906 exmidomniim 7084 exmidfodomrlemim 7136 exmidontriimlem1 7156 |
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