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 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 1418	. . . . . . . . . . . . . 14 ⊢ Ⅎv(f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒))
2	1	sb8eu 1910	. . . . . . . . . . . . 13 ⊢ (∃!f(f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)) ↔ ∃!v[v / f](f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)))
3		eleq12 2099	. . . . . . . . . . . . . . . . . . . 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 923	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 = z ∧ f = v ∧ 𝑏 = y) → (f ∈ 𝑐 ↔ v ∈ z))
6		eleq12 2099	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑐 = z ∧ 𝑒 = u) → (𝑐 ∈ 𝑒 ↔ z ∈ u))
7	6	3ad2antl1 1065	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑐 = z ∧ f = v ∧ 𝑏 = y) ∧ 𝑒 = u) → (𝑐 ∈ 𝑒 ↔ z ∈ u))
8		eleq12 2099	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((f = v ∧ 𝑒 = u) → (f ∈ 𝑒 ↔ v ∈ u))
9	8	3ad2antl2 1066	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑐 = z ∧ f = v ∧ 𝑏 = y) ∧ 𝑒 = u) → (f ∈ 𝑒 ↔ v ∈ u))
10	7, 9	anbi12d 442	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑐 = z ∧ f = v ∧ 𝑏 = y) ∧ 𝑒 = u) → ((𝑐 ∈ 𝑒 ∧ f ∈ 𝑒) ↔ (z ∈ u ∧ v ∈ u)))
11		simpl3 908	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑐 = z ∧ f = v ∧ 𝑏 = y) ∧ 𝑒 = u) → 𝑏 = y)
12	10, 11	cbvrexdva2 2532	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 = z ∧ f = v ∧ 𝑏 = y) → (∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒) ↔ ∃u ∈ y (z ∈ u ∧ v ∈ u)))
13	5, 12	anbi12d 442	. . . . . . . . . . . . . . . . 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 1109	. . . . . . . . . . . . . . . 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 1103	. . . . . . . . . . . . . . 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 1669	. . . . . . . . . . . . . 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 1905	. . . . . . . . . . . . 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 2307	. . . . . . . . . . . 12 ⊢ (∃!f ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒) ↔ ∃!f(f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)))
20		df-reu 2307	. . . . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ (((𝑐 = z ∧ 𝑏 = y) ∧ 𝑑 = w) → 𝑐 = z)
24	22, 23	cbvraldva2 2531	. . . . . . . . 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 445	. . . . . . 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 481	. . . . . . 7 ⊢ (((𝑎 = x ∧ 𝑏 = y) ∧ 𝑐 = z) → 𝑎 = x)
28	26, 27	cbvraldva2 2531	. . . . . 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 1801	. . . . 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 1791	. . . 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 1339	. . 3 ⊢ ∀𝑎∃𝑏∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑐 ∃!f ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)
33	32	spi 1426	. 2 ⊢ ∃𝑏∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑐 ∃!f ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)
34	33	acexmidlemv 5453	1 ⊢ (φ ∨ ¬ φ)

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)