Theorem acexmid 5433

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	⊢ ∃y∀z ∈ x ∀w ∈ z ∃!v ∈ z ∃u ∈ y (z ∈ u ∧ v ∈ u)

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.

Step	Hyp	Ref	Expression
1		nfv 1403	. . . . . . . . . . . . . 14 ⊢ Ⅎv(f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒))
2	1	sb8eu 1896	. . . . . . . . . . . . 13 ⊢ (∃!f(f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)) ↔ ∃!v[v / f](f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)))
3		eleq12 2085	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((f = v ∧ 𝑐 = z) → (f ∈ 𝑐 ↔ v ∈ z))
4	3	ancoms 255	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑐 = z ∧ f = v) → (f ∈ 𝑐 ↔ v ∈ z))
5	4	3adant3 912	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 = z ∧ f = v ∧ 𝑏 = y) → (f ∈ 𝑐 ↔ v ∈ z))
6		eleq12 2085	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑐 = z ∧ 𝑒 = u) → (𝑐 ∈ 𝑒 ↔ z ∈ u))
7	6	3ad2antl1 1054	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑐 = z ∧ f = v ∧ 𝑏 = y) ∧ 𝑒 = u) → (𝑐 ∈ 𝑒 ↔ z ∈ u))
8		eleq12 2085	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((f = v ∧ 𝑒 = u) → (f ∈ 𝑒 ↔ v ∈ u))
9	8	3ad2antl2 1055	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑐 = z ∧ f = v ∧ 𝑏 = y) ∧ 𝑒 = u) → (f ∈ 𝑒 ↔ v ∈ u))
10	7, 9	anbi12d 445	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑐 = z ∧ f = v ∧ 𝑏 = y) ∧ 𝑒 = u) → ((𝑐 ∈ 𝑒 ∧ f ∈ 𝑒) ↔ (z ∈ u ∧ v ∈ u)))
11		simpl3 897	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑐 = z ∧ f = v ∧ 𝑏 = y) ∧ 𝑒 = u) → 𝑏 = y)
12	10, 11	cbvrexdva2 2515	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 = z ∧ f = v ∧ 𝑏 = y) → (∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒) ↔ ∃u ∈ y (z ∈ u ∧ v ∈ u)))
13	5, 12	anbi12d 445	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 = z ∧ f = v ∧ 𝑏 = y) → ((f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)) ↔ (v ∈ z ∧ ∃u ∈ y (z ∈ u ∧ v ∈ u))))
14	13	3com23 1096	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 = z ∧ 𝑏 = y ∧ f = v) → ((f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)) ↔ (v ∈ z ∧ ∃u ∈ y (z ∈ u ∧ v ∈ u))))
15	14	3expa 1090	. . . . . . . . . . . . . . 15 ⊢ (((𝑐 = z ∧ 𝑏 = y) ∧ f = v) → ((f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)) ↔ (v ∈ z ∧ ∃u ∈ y (z ∈ u ∧ v ∈ u))))
16	15	sbiedv 1655	. . . . . . . . . . . . . 14 ⊢ ((𝑐 = z ∧ 𝑏 = y) → ([v / f](f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)) ↔ (v ∈ z ∧ ∃u ∈ y (z ∈ u ∧ v ∈ u))))
17	16	eubidv 1891	. . . . . . . . . . . . 13 ⊢ ((𝑐 = z ∧ 𝑏 = y) → (∃!v[v / f](f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)) ↔ ∃!v(v ∈ z ∧ ∃u ∈ y (z ∈ u ∧ v ∈ u))))
18	2, 17	syl5bb 181	. . . . . . . . . . . 12 ⊢ ((𝑐 = z ∧ 𝑏 = y) → (∃!f(f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)) ↔ ∃!v(v ∈ z ∧ ∃u ∈ y (z ∈ u ∧ v ∈ u))))
19		df-reu 2290	. . . . . . . . . . . 12 ⊢ (∃!f ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒) ↔ ∃!f(f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)))
20		df-reu 2290	. . . . . . . . . . . 12 ⊢ (∃!v ∈ z ∃u ∈ y (z ∈ u ∧ v ∈ u) ↔ ∃!v(v ∈ z ∧ ∃u ∈ y (z ∈ u ∧ v ∈ u)))
21	18, 19, 20	3bitr4g 212	. . . . . . . . . . 11 ⊢ ((𝑐 = z ∧ 𝑏 = y) → (∃!f ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒) ↔ ∃!v ∈ z ∃u ∈ y (z ∈ u ∧ v ∈ u)))
22	21	adantr 261	. . . . . . . . . 10 ⊢ (((𝑐 = z ∧ 𝑏 = y) ∧ 𝑑 = w) → (∃!f ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒) ↔ ∃!v ∈ z ∃u ∈ y (z ∈ u ∧ v ∈ u)))
23		simpll 469	. . . . . . . . . 10 ⊢ (((𝑐 = z ∧ 𝑏 = y) ∧ 𝑑 = w) → 𝑐 = z)
24	22, 23	cbvraldva2 2514	. . . . . . . . 9 ⊢ ((𝑐 = z ∧ 𝑏 = y) → (∀𝑑 ∈ 𝑐 ∃!f ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒) ↔ ∀w ∈ z ∃!v ∈ z ∃u ∈ y (z ∈ u ∧ v ∈ u)))
25	24	ancoms 255	. . . . . . . 8 ⊢ ((𝑏 = y ∧ 𝑐 = z) → (∀𝑑 ∈ 𝑐 ∃!f ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒) ↔ ∀w ∈ z ∃!v ∈ z ∃u ∈ y (z ∈ u ∧ v ∈ u)))
26	25	adantll 448	. . . . . . 7 ⊢ (((𝑎 = x ∧ 𝑏 = y) ∧ 𝑐 = z) → (∀𝑑 ∈ 𝑐 ∃!f ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒) ↔ ∀w ∈ z ∃!v ∈ z ∃u ∈ y (z ∈ u ∧ v ∈ u)))
27		simpll 469	. . . . . . 7 ⊢ (((𝑎 = x ∧ 𝑏 = y) ∧ 𝑐 = z) → 𝑎 = x)
28	26, 27	cbvraldva2 2514	. . . . . 6 ⊢ ((𝑎 = x ∧ 𝑏 = y) → (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑐 ∃!f ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒) ↔ ∀z ∈ x ∀w ∈ z ∃!v ∈ z ∃u ∈ y (z ∈ u ∧ v ∈ u)))
29	28	cbvexdva 1787	. . . . 5 ⊢ (𝑎 = x → (∃𝑏∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑐 ∃!f ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒) ↔ ∃y∀z ∈ x ∀w ∈ z ∃!v ∈ z ∃u ∈ y (z ∈ u ∧ v ∈ u)))
30	29	cbvalv 1777	. . . 4 ⊢ (∀𝑎∃𝑏∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑐 ∃!f ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒) ↔ ∀x∃y∀z ∈ x ∀w ∈ z ∃!v ∈ z ∃u ∈ y (z ∈ u ∧ v ∈ u))
31		acexmid.choice	. . . 4 ⊢ ∃y∀z ∈ x ∀w ∈ z ∃!v ∈ z ∃u ∈ y (z ∈ u ∧ v ∈ u)
32	30, 31	mpgbir 1322	. . 3 ⊢ ∀𝑎∃𝑏∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑐 ∃!f ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)
33	32	spi 1412	. 2 ⊢ ∃𝑏∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑐 ∃!f ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)
34	33	acexmidlemv 5432	1 ⊢ (φ ∨ ¬ φ)

Syntax hints:  ¬ wn 3   ∧ wa 97   ↔ wb 98   ∨ wo 616   ∧ w3a 873  ∀wal 1226  ∃wex 1363  [wsb 1628  ∃!weu 1883  ∀wral 2283  ∃wrex 2284  ∃!wreu 2285

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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-nul 3856  ax-pow 3900  ax-pr 3917

This theorem depends on definitions:  df-bi 110  df-3or 874  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-reu 2290  df-rab 2292  df-v 2536  df-sbc 2741  df-dif 2896  df-un 2898  df-in 2900  df-ss 2907  df-nul 3201  df-pw 3335  df-sn 3355  df-pr 3356  df-uni 3554  df-tr 3828  df-iord 4050  df-on 4052  df-suc 4055  df-iota 4792  df-riota 5391

This theorem is referenced by: (None)