Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  regexmid Unicode version

Theorem regexmid 4260
 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.)
Hypothesis
Ref Expression
regexmid.1
Assertion
Ref Expression
regexmid
Distinct variable group:   ,,,

Proof of Theorem regexmid
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqid 2040 . . 3
21regexmidlemm 4257 . 2
3 pp0ex 3940 . . . 4
43rabex 3901 . . 3
5 eleq2 2101 . . . . 5
65exbidv 1706 . . . 4
7 eleq2 2101 . . . . . . . . 9
87notbid 592 . . . . . . . 8
98imbi2d 219 . . . . . . 7
109albidv 1705 . . . . . 6
115, 10anbi12d 442 . . . . 5
1211exbidv 1706 . . . 4
136, 12imbi12d 223 . . 3
14 regexmid.1 . . 3
154, 13, 14vtocl 2608 . 2
161regexmidlem1 4258 . 2
172, 15, 16mp2b 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)
 Copyright terms: Public domain W3C validator