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Theorem acexmid 5813
 Description: The axiom of choice implies excluded middle. Theorem 1.3 in [Bauer] p. 483. The statement of the axiom of choice given here is ac2 in the Metamath Proof Explorer (version of 3-Aug-2019). In particular, note that the choice function provides a value when is inhabited (as opposed to nonempty as in some statements of the axiom of choice). Essentially the same proof can also be found at "The axiom of choice implies instances of EM", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". Often referred to as Diaconescu's theorem, or Diaconescu-Goodman-Myhill theorem, after Radu Diaconescu who discovered it in 1975 in the framework of topos theory and N. D. Goodman and John Myhill in 1978 in the framework of set theory (although it already appeared as an exercise in Errett Bishop's book Foundations of Constructive Analysis from 1967). For this theorem stated using the df-ac 7120 and df-exmid 4151 syntaxes, see exmidac 7123. (Contributed by Jim Kingdon, 4-Aug-2019.)
Hypothesis
Ref Expression
acexmid.choice
Assertion
Ref Expression
acexmid
Distinct variable group:   ,,,,,
Allowed substitution hints:   (,,,,,)

Proof of Theorem acexmid
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1505 . . . . . . . . . . . . . 14
21sb8eu 2016 . . . . . . . . . . . . 13
3 eleq12 2219 . . . . . . . . . . . . . . . . . . . 20
43ancoms 266 . . . . . . . . . . . . . . . . . . 19
543adant3 1002 . . . . . . . . . . . . . . . . . 18
6 eleq12 2219 . . . . . . . . . . . . . . . . . . . . 21
763ad2antl1 1144 . . . . . . . . . . . . . . . . . . . 20
8 eleq12 2219 . . . . . . . . . . . . . . . . . . . . 21
983ad2antl2 1145 . . . . . . . . . . . . . . . . . . . 20
107, 9anbi12d 465 . . . . . . . . . . . . . . . . . . 19
11 simpl3 987 . . . . . . . . . . . . . . . . . . 19
1210, 11cbvrexdva2 2685 . . . . . . . . . . . . . . . . . 18
135, 12anbi12d 465 . . . . . . . . . . . . . . . . 17
14133com23 1188 . . . . . . . . . . . . . . . 16
15143expa 1182 . . . . . . . . . . . . . . 15
1615sbiedv 1766 . . . . . . . . . . . . . 14
1716eubidv 2011 . . . . . . . . . . . . 13
182, 17syl5bb 191 . . . . . . . . . . . 12
19 df-reu 2439 . . . . . . . . . . . 12
20 df-reu 2439 . . . . . . . . . . . 12
2118, 19, 203bitr4g 222 . . . . . . . . . . 11
2221adantr 274 . . . . . . . . . 10
23 simpll 519 . . . . . . . . . 10
2422, 23cbvraldva2 2684 . . . . . . . . 9
2524ancoms 266 . . . . . . . 8
2625adantll 468 . . . . . . 7
27 simpll 519 . . . . . . 7
2826, 27cbvraldva2 2684 . . . . . 6
2928cbvexdva 1906 . . . . 5
3029cbvalv 1894 . . . 4
31 acexmid.choice . . . 4
3230, 31mpgbir 1430 . . 3
3332spi 1513 . 2
3433acexmidlemv 5812 1
 Colors of variables: wff set class Syntax hints:   wn 3   wa 103   wb 104   wo 698   w3a 963  wal 1330  wex 1469  wsb 1739  weu 2003  wral 2432  wrex 2433  wreu 2434 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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164 This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-pr 3563  df-uni 3769  df-tr 4059  df-iord 4321  df-on 4323  df-suc 4326  df-iota 5128  df-riota 5770 This theorem is referenced by: (None)
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