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Theorem acexmid 5539
 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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1437 . . . . . . . . . . . . . 14
21sb8eu 1929 . . . . . . . . . . . . 13
3 eleq12 2118 . . . . . . . . . . . . . . . . . . . 20
43ancoms 259 . . . . . . . . . . . . . . . . . . 19
543adant3 935 . . . . . . . . . . . . . . . . . 18
6 eleq12 2118 . . . . . . . . . . . . . . . . . . . . 21
763ad2antl1 1077 . . . . . . . . . . . . . . . . . . . 20
8 eleq12 2118 . . . . . . . . . . . . . . . . . . . . 21
983ad2antl2 1078 . . . . . . . . . . . . . . . . . . . 20
107, 9anbi12d 450 . . . . . . . . . . . . . . . . . . 19
11 simpl3 920 . . . . . . . . . . . . . . . . . . 19
1210, 11cbvrexdva2 2555 . . . . . . . . . . . . . . . . . 18
135, 12anbi12d 450 . . . . . . . . . . . . . . . . 17
14133com23 1121 . . . . . . . . . . . . . . . 16
15143expa 1115 . . . . . . . . . . . . . . 15
1615sbiedv 1688 . . . . . . . . . . . . . 14
1716eubidv 1924 . . . . . . . . . . . . 13
182, 17syl5bb 185 . . . . . . . . . . . 12
19 df-reu 2330 . . . . . . . . . . . 12
20 df-reu 2330 . . . . . . . . . . . 12
2118, 19, 203bitr4g 216 . . . . . . . . . . 11
2221adantr 265 . . . . . . . . . 10
23 simpll 489 . . . . . . . . . 10
2422, 23cbvraldva2 2554 . . . . . . . . 9
2524ancoms 259 . . . . . . . 8
2625adantll 453 . . . . . . 7
27 simpll 489 . . . . . . 7
2826, 27cbvraldva2 2554 . . . . . 6
2928cbvexdva 1820 . . . . 5
3029cbvalv 1810 . . . 4
31 acexmid.choice . . . 4
3230, 31mpgbir 1358 . . 3
3332spi 1445 . 2
3433acexmidlemv 5538 1
 Colors of variables: wff set class Syntax hints:   wn 3   wa 101   wb 102   wo 639   w3a 896  wal 1257  wex 1397  wsb 1661  weu 1916  wral 2323  wrex 2324  wreu 2325 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972 This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133  df-suc 4136  df-iota 4895  df-riota 5496 This theorem is referenced by: (None)
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