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 1402	. . . . . . . . . . . . . 14 ⊢ Ⅎv(f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒))
2	1	sb8eu 1895	. . . . . . . . . . . . 13 ⊢ (∃!f(f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)) ↔ ∃!v[v / f](f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)))
3		eleq12 2084	. . . . . . . . . . . . . . . . . . . 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 2084	. . . . . . . . . . . . . . . . . . . . 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 2084	. . . . . . . . . . . . . . . . . . . . 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 2514	. . . . . . . . . . . . . . . . . 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 1654	. . . . . . . . . . . . . 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 1890	. . . . . . . . . . . . 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 2289	. . . . . . . . . . . 12 ⊢ (∃!f ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒) ↔ ∃!f(f ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ f ∈ 𝑒)))
20		df-reu 2289	. . . . . . . . . . . 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 2513	. . . . . . . . 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 2513	. . . . . 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 1786	. . . . 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 1776	. . . 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 1411	. 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 1362  [wsb 1627  ∃!weu 1882  ∀wral 2282  ∃wrex 2283  ∃!wreu 2284

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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3847  ax-nul 3855  ax-pow 3899  ax-pr 3916

This theorem depends on definitions:  df-bi 110  df-3or 874  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-reu 2289  df-rab 2291  df-v 2535  df-sbc 2740  df-dif 2895  df-un 2897  df-in 2899  df-ss 2906  df-nul 3200  df-pw 3334  df-sn 3354  df-pr 3355  df-uni 3553  df-tr 3827  df-iord 4050  df-on 4052  df-suc 4055  df-iota 4792  df-riota 5391

This theorem is referenced by: (None)