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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 3187 | . . 3 | |
2 | df-exmid 4127 | . . . 4 EXMID DECID | |
3 | p0ex 4120 | . . . . . 6 | |
4 | 3 | rabex 4080 | . . . . 5 |
5 | sseq1 3125 | . . . . . 6 | |
6 | eleq2 2204 | . . . . . . 7 | |
7 | 6 | dcbid 824 | . . . . . 6 DECID DECID |
8 | 5, 7 | imbi12d 233 | . . . . 5 DECID DECID |
9 | 4, 8 | spcv 2783 | . . . 4 DECID DECID |
10 | 2, 9 | sylbi 120 | . . 3 EXMID DECID |
11 | 1, 10 | mpi 15 | . 2 EXMID DECID |
12 | 0ex 4063 | . . . . 5 | |
13 | 12 | snid 3563 | . . . 4 |
14 | biidd 171 | . . . . 5 | |
15 | 14 | elrab 2844 | . . . 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 1330 wceq 1332 wcel 1481 crab 2421 wss 3076 c0 3368 csn 3532 EXMIDwem 4126 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rab 2426 df-v 2691 df-dif 3078 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-exmid 4127 |
This theorem is referenced by: exmidn0m 4132 exmid0el 4135 exmidel 4136 exmidundif 4137 exmidundifim 4138 sbthlemi3 6855 sbthlemi5 6857 sbthlemi6 6858 exmidomniim 7021 exmidfodomrlemim 7074 |
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