ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  acexmid GIF version

Theorem acexmid 5593
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 𝑦 provides a value when 𝑧 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 𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)
Assertion
Ref Expression
acexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem acexmid
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1464 . . . . . . . . . . . . . 14 𝑣(𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒))
21sb8eu 1958 . . . . . . . . . . . . 13 (∃!𝑓(𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ ∃!𝑣[𝑣 / 𝑓](𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)))
3 eleq12 2149 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 = 𝑣𝑐 = 𝑧) → (𝑓𝑐𝑣𝑧))
43ancoms 264 . . . . . . . . . . . . . . . . . . 19 ((𝑐 = 𝑧𝑓 = 𝑣) → (𝑓𝑐𝑣𝑧))
543adant3 961 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) → (𝑓𝑐𝑣𝑧))
6 eleq12 2149 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐 = 𝑧𝑒 = 𝑢) → (𝑐𝑒𝑧𝑢))
763ad2antl1 1103 . . . . . . . . . . . . . . . . . . . 20 (((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → (𝑐𝑒𝑧𝑢))
8 eleq12 2149 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 = 𝑣𝑒 = 𝑢) → (𝑓𝑒𝑣𝑢))
983ad2antl2 1104 . . . . . . . . . . . . . . . . . . . 20 (((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → (𝑓𝑒𝑣𝑢))
107, 9anbi12d 457 . . . . . . . . . . . . . . . . . . 19 (((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → ((𝑐𝑒𝑓𝑒) ↔ (𝑧𝑢𝑣𝑢)))
11 simpl3 946 . . . . . . . . . . . . . . . . . . 19 (((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → 𝑏 = 𝑦)
1210, 11cbvrexdva2 2590 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) → (∃𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢)))
135, 12anbi12d 457 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) → ((𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ (𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
14133com23 1147 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝑧𝑏 = 𝑦𝑓 = 𝑣) → ((𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ (𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
15143expa 1141 . . . . . . . . . . . . . . 15 (((𝑐 = 𝑧𝑏 = 𝑦) ∧ 𝑓 = 𝑣) → ((𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ (𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
1615sbiedv 1716 . . . . . . . . . . . . . 14 ((𝑐 = 𝑧𝑏 = 𝑦) → ([𝑣 / 𝑓](𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ (𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
1716eubidv 1953 . . . . . . . . . . . . 13 ((𝑐 = 𝑧𝑏 = 𝑦) → (∃!𝑣[𝑣 / 𝑓](𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ ∃!𝑣(𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
182, 17syl5bb 190 . . . . . . . . . . . 12 ((𝑐 = 𝑧𝑏 = 𝑦) → (∃!𝑓(𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ ∃!𝑣(𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
19 df-reu 2362 . . . . . . . . . . . 12 (∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∃!𝑓(𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)))
20 df-reu 2362 . . . . . . . . . . . 12 (∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃!𝑣(𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2118, 19, 203bitr4g 221 . . . . . . . . . . 11 ((𝑐 = 𝑧𝑏 = 𝑦) → (∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2221adantr 270 . . . . . . . . . 10 (((𝑐 = 𝑧𝑏 = 𝑦) ∧ 𝑑 = 𝑤) → (∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
23 simpll 496 . . . . . . . . . 10 (((𝑐 = 𝑧𝑏 = 𝑦) ∧ 𝑑 = 𝑤) → 𝑐 = 𝑧)
2422, 23cbvraldva2 2589 . . . . . . . . 9 ((𝑐 = 𝑧𝑏 = 𝑦) → (∀𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2524ancoms 264 . . . . . . . 8 ((𝑏 = 𝑦𝑐 = 𝑧) → (∀𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2625adantll 460 . . . . . . 7 (((𝑎 = 𝑥𝑏 = 𝑦) ∧ 𝑐 = 𝑧) → (∀𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
27 simpll 496 . . . . . . 7 (((𝑎 = 𝑥𝑏 = 𝑦) ∧ 𝑐 = 𝑧) → 𝑎 = 𝑥)
2826, 27cbvraldva2 2589 . . . . . 6 ((𝑎 = 𝑥𝑏 = 𝑦) → (∀𝑐𝑎𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∀𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2928cbvexdva 1849 . . . . 5 (𝑎 = 𝑥 → (∃𝑏𝑐𝑎𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
3029cbvalv 1839 . . . 4 (∀𝑎𝑏𝑐𝑎𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∀𝑥𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
31 acexmid.choice . . . 4 𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)
3230, 31mpgbir 1385 . . 3 𝑎𝑏𝑐𝑎𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒)
3332spi 1472 . 2 𝑏𝑐𝑎𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒)
3433acexmidlemv 5592 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wb 103  wo 662  w3a 922  wal 1285  wex 1424  [wsb 1689  ∃!weu 1945  wral 2355  wrex 2356  ∃!wreu 2357
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003
This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-uni 3631  df-tr 3905  df-iord 4160  df-on 4162  df-suc 4165  df-iota 4937  df-riota 5550
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator