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Theorem acexmid 5454
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 non-empty 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).

(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  a  b  c  d  e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . . . . . . . . . . . 14  F/  c  e  b  c  e  e
21sb8eu 1910 . . . . . . . . . . . . 13  c  e  b  c  e  e  c  e  b  c  e  e
3 eleq12 2099 . . . . . . . . . . . . . . . . . . . 20  c  c
43ancoms 255 . . . . . . . . . . . . . . . . . . 19  c  c
543adant3 923 . . . . . . . . . . . . . . . . . 18  c  b  c
6 eleq12 2099 . . . . . . . . . . . . . . . . . . . . 21  c  e  c  e
763ad2antl1 1065 . . . . . . . . . . . . . . . . . . . 20  c  b  e  c  e
8 eleq12 2099 . . . . . . . . . . . . . . . . . . . . 21  e  e
983ad2antl2 1066 . . . . . . . . . . . . . . . . . . . 20  c  b  e  e
107, 9anbi12d 442 . . . . . . . . . . . . . . . . . . 19  c  b  e  c  e  e
11 simpl3 908 . . . . . . . . . . . . . . . . . . 19  c  b  e  b
1210, 11cbvrexdva2 2532 . . . . . . . . . . . . . . . . . 18  c  b  e  b  c  e  e
135, 12anbi12d 442 . . . . . . . . . . . . . . . . 17  c  b  c  e  b  c  e  e
14133com23 1109 . . . . . . . . . . . . . . . 16  c  b  c  e  b  c  e  e
15143expa 1103 . . . . . . . . . . . . . . 15  c  b  c  e  b  c  e  e
1615sbiedv 1669 . . . . . . . . . . . . . 14  c  b  c  e  b  c  e  e
1716eubidv 1905 . . . . . . . . . . . . 13  c  b  c  e  b  c  e  e
182, 17syl5bb 181 . . . . . . . . . . . 12  c  b  c  e  b  c  e  e
19 df-reu 2307 . . . . . . . . . . . 12  c  e  b 
c  e  e  c  e  b  c  e  e
20 df-reu 2307 . . . . . . . . . . . 12
2118, 19, 203bitr4g 212 . . . . . . . . . . 11  c  b  c  e  b  c  e  e
2221adantr 261 . . . . . . . . . 10  c  b  d  c  e  b 
c  e  e
23 simpll 481 . . . . . . . . . 10  c  b  d  c
2422, 23cbvraldva2 2531 . . . . . . . . 9  c  b  d  c  c  e  b  c  e  e
2524ancoms 255 . . . . . . . 8  b  c  d  c  c  e  b  c  e  e
2625adantll 445 . . . . . . 7  a  b  c  d  c  c  e  b 
c  e  e
27 simpll 481 . . . . . . 7  a  b  c  a
2826, 27cbvraldva2 2531 . . . . . 6  a  b  c  a  d  c  c  e  b  c  e  e
2928cbvexdva 1801 . . . . 5  a  b c  a  d  c  c  e  b  c  e  e
3029cbvalv 1791 . . . 4  a b c  a  d  c  c  e  b  c  e  e
31 acexmid.choice . . . 4
3230, 31mpgbir 1339 . . 3  a b c  a  d  c  c  e  b  c  e  e
3332spi 1426 . 2  b c  a  d  c  c  e  b  c  e  e
3433acexmidlemv 5453 1
Colors of variables: wff set class
Syntax hints:   wn 3   wa 97   wb 98   wo 628   w3a 884  wal 1240  wex 1378  wsb 1642  weu 1897  wral 2300  wrex 2301  wreu 2302
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074  df-iota 4810  df-riota 5411
This theorem is referenced by: (None)
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