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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 828, peircedc 899, or condc 838. (Contributed by Jim Kingdon, 18-Jun-2022.) |
Ref | Expression |
---|---|
exmidexmid | EXMID DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3177 | . . 3 | |
2 | df-exmid 4114 | . . . 4 EXMID DECID | |
3 | p0ex 4107 | . . . . . 6 | |
4 | 3 | rabex 4067 | . . . . 5 |
5 | sseq1 3115 | . . . . . 6 | |
6 | eleq2 2201 | . . . . . . 7 | |
7 | 6 | dcbid 823 | . . . . . 6 DECID DECID |
8 | 5, 7 | imbi12d 233 | . . . . 5 DECID DECID |
9 | 4, 8 | spcv 2774 | . . . 4 DECID DECID |
10 | 2, 9 | sylbi 120 | . . 3 EXMID DECID |
11 | 1, 10 | mpi 15 | . 2 EXMID DECID |
12 | 0ex 4050 | . . . . 5 | |
13 | 12 | snid 3551 | . . . 4 |
14 | biidd 171 | . . . . 5 | |
15 | 14 | elrab 2835 | . . . 4 |
16 | 13, 15 | mpbiran 924 | . . 3 |
17 | 16 | dcbii 825 | . 2 DECID DECID |
18 | 11, 17 | sylib 121 | 1 EXMID DECID |
Colors of variables: wff set class |
Syntax hints: wi 4 DECID wdc 819 wal 1329 wceq 1331 wcel 1480 crab 2418 wss 3066 c0 3358 csn 3522 EXMIDwem 4113 |
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 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rab 2423 df-v 2683 df-dif 3068 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-exmid 4114 |
This theorem is referenced by: exmidn0m 4119 exmid0el 4122 exmidel 4123 exmidundif 4124 exmidundifim 4125 sbthlemi3 6840 sbthlemi5 6842 sbthlemi6 6843 exmidomniim 7006 exmidfodomrlemim 7050 |
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