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Theorem acexmid 5431
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 1398 . . . . . . . . . . . . . 14  F/  c  e  b  c  e  e
21sb8eu 1891 . . . . . . . . . . . . 13  c  e  b  c  e  e  c  e  b  c  e  e
3 eleq12 2080 . . . . . . . . . . . . . . . . . . . 20  c  c
43ancoms 255 . . . . . . . . . . . . . . . . . . 19  c  c
543adant3 910 . . . . . . . . . . . . . . . . . 18  c  b  c
6 eleq12 2080 . . . . . . . . . . . . . . . . . . . . 21  c  e  c  e
763ad2antl1 1052 . . . . . . . . . . . . . . . . . . . 20  c  b  e  c  e
8 eleq12 2080 . . . . . . . . . . . . . . . . . . . . 21  e  e
983ad2antl2 1053 . . . . . . . . . . . . . . . . . . . 20  c  b  e  e
107, 9anbi12d 445 . . . . . . . . . . . . . . . . . . 19  c  b  e  c  e  e
11 simpl3 895 . . . . . . . . . . . . . . . . . . 19  c  b  e  b
1210, 11cbvrexdva2 2512 . . . . . . . . . . . . . . . . . 18  c  b  e  b  c  e  e
135, 12anbi12d 445 . . . . . . . . . . . . . . . . 17  c  b  c  e  b  c  e  e
14133com23 1094 . . . . . . . . . . . . . . . 16  c  b  c  e  b  c  e  e
15143expa 1088 . . . . . . . . . . . . . . 15  c  b  c  e  b  c  e  e
1615sbiedv 1650 . . . . . . . . . . . . . 14  c  b  c  e  b  c  e  e
1716eubidv 1886 . . . . . . . . . . . . 13  c  b  c  e  b  c  e  e
182, 17syl5bb 181 . . . . . . . . . . . 12  c  b  c  e  b  c  e  e
19 df-reu 2287 . . . . . . . . . . . 12  c  e  b 
c  e  e  c  e  b  c  e  e
20 df-reu 2287 . . . . . . . . . . . 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 469 . . . . . . . . . 10  c  b  d  c
2422, 23cbvraldva2 2511 . . . . . . . . 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 448 . . . . . . 7  a  b  c  d  c  c  e  b 
c  e  e
27 simpll 469 . . . . . . 7  a  b  c  a
2826, 27cbvraldva2 2511 . . . . . 6  a  b  c  a  d  c  c  e  b  c  e  e
2928cbvexdva 1782 . . . . 5  a  b c  a  d  c  c  e  b  c  e  e
3029cbvalv 1772 . . . 4  a b c  a  d  c  c  e  b  c  e  e
31 acexmid.choice . . . 4
3230, 31mpgbir 1318 . . 3  a b c  a  d  c  c  e  b  c  e  e
3332spi 1407 . 2  b c  a  d  c  c  e  b  c  e  e
3433acexmidlemv 5430 1
Colors of variables: wff set class
Syntax hints:   wn 3   wa 97   wb 98   wo 616   w3a 871  wal 1224  wex 1358  wsb 1623  weu 1878  wral 2280  wrex 2281  wreu 2282
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3or 872  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-uni 3551  df-tr 3825  df-iord 4048  df-on 4050  df-suc 4053  df-iota 4790  df-riota 5389
This theorem is referenced by: (None)
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