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Theorem regexmid 4492
 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 4494. (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 2157 . . 3
21regexmidlemm 4489 . 2
3 pp0ex 4149 . . . 4
43rabex 4108 . . 3
5 eleq2 2221 . . . . 5
65exbidv 1805 . . . 4
7 eleq2 2221 . . . . . . . . 9
87notbid 657 . . . . . . . 8
98imbi2d 229 . . . . . . 7
109albidv 1804 . . . . . 6
115, 10anbi12d 465 . . . . 5
1211exbidv 1805 . . . 4
136, 12imbi12d 233 . . 3
14 regexmid.1 . . 3
154, 13, 14vtocl 2766 . 2
161regexmidlem1 4490 . 2
172, 15, 16mp2b 8 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wo 698  wal 1333   wceq 1335  wex 1472   wcel 2128  crab 2439  c0 3394  csn 3560  cpr 3561 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567 This theorem is referenced by: (None)
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