Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > regexmid | Unicode version |
Description: The axiom of foundation
implies excluded middle.
By foundation (or regularity), we mean the principle that every inhabited set has an element which is minimal (when arranged by ). The statement of foundation here is taken from Metamath Proof Explorer's ax-reg, and is identical (modulo one unnecessary quantifier) to the statement of foundation in Theorem "Foundation implies instances of EM" of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". For this reason, IZF does not adopt foundation as an axiom and instead replaces it with ax-setind 4460. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Ref | Expression |
---|---|
regexmid.1 |
Ref | Expression |
---|---|
regexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2140 | . . 3 | |
2 | 1 | regexmidlemm 4455 | . 2 |
3 | pp0ex 4121 | . . . 4 | |
4 | 3 | rabex 4080 | . . 3 |
5 | eleq2 2204 | . . . . 5 | |
6 | 5 | exbidv 1798 | . . . 4 |
7 | eleq2 2204 | . . . . . . . . 9 | |
8 | 7 | notbid 657 | . . . . . . . 8 |
9 | 8 | imbi2d 229 | . . . . . . 7 |
10 | 9 | albidv 1797 | . . . . . 6 |
11 | 5, 10 | anbi12d 465 | . . . . 5 |
12 | 11 | exbidv 1798 | . . . 4 |
13 | 6, 12 | imbi12d 233 | . . 3 |
14 | regexmid.1 | . . 3 | |
15 | 4, 13, 14 | vtocl 2743 | . 2 |
16 | 1 | regexmidlem1 4456 | . 2 |
17 | 2, 15, 16 | mp2b 8 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 wal 1330 wceq 1332 wex 1469 wcel 1481 crab 2421 c0 3368 csn 3532 cpr 3533 |
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-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-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |