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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 4262. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Ref | Expression |
---|---|
regexmid.1 |
Ref | Expression |
---|---|
regexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2040 | . . 3 | |
2 | 1 | regexmidlemm 4257 | . 2 |
3 | pp0ex 3940 | . . . 4 | |
4 | 3 | rabex 3901 | . . 3 |
5 | eleq2 2101 | . . . . 5 | |
6 | 5 | exbidv 1706 | . . . 4 |
7 | eleq2 2101 | . . . . . . . . 9 | |
8 | 7 | notbid 592 | . . . . . . . 8 |
9 | 8 | imbi2d 219 | . . . . . . 7 |
10 | 9 | albidv 1705 | . . . . . 6 |
11 | 5, 10 | anbi12d 442 | . . . . 5 |
12 | 11 | exbidv 1706 | . . . 4 |
13 | 6, 12 | imbi12d 223 | . . 3 |
14 | regexmid.1 | . . 3 | |
15 | 4, 13, 14 | vtocl 2608 | . 2 |
16 | 1 | regexmidlem1 4258 | . 2 |
17 | 2, 15, 16 | mp2b 8 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wo 629 wal 1241 wceq 1243 wex 1381 wcel 1393 crab 2310 c0 3224 csn 3375 cpr 3376 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 |
This theorem is referenced by: (None) |
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