Theorem acexmid 5890

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 nonempty 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).

For this theorem stated using the df-ac 7223 and df-exmid 4210 syntaxes, see exmidac 7226. (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 1539	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑣(𝑓 ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒))
2	1	sb8eu 2051	. . . . . . . . . . . . 13 ⊢ (∃!𝑓(𝑓 ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒)) ↔ ∃!𝑣[𝑣 / 𝑓](𝑓 ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒)))
3		eleq12 2254	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 = 𝑣 ∧ 𝑐 = 𝑧) → (𝑓 ∈ 𝑐 ↔ 𝑣 ∈ 𝑧))
4	3	ancoms 268	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑐 = 𝑧 ∧ 𝑓 = 𝑣) → (𝑓 ∈ 𝑐 ↔ 𝑣 ∈ 𝑧))
5	4	3adant3 1019	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 = 𝑧 ∧ 𝑓 = 𝑣 ∧ 𝑏 = 𝑦) → (𝑓 ∈ 𝑐 ↔ 𝑣 ∈ 𝑧))
6		eleq12 2254	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑐 = 𝑧 ∧ 𝑒 = 𝑢) → (𝑐 ∈ 𝑒 ↔ 𝑧 ∈ 𝑢))
7	6	3ad2antl1 1161	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑐 = 𝑧 ∧ 𝑓 = 𝑣 ∧ 𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → (𝑐 ∈ 𝑒 ↔ 𝑧 ∈ 𝑢))
8		eleq12 2254	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 = 𝑣 ∧ 𝑒 = 𝑢) → (𝑓 ∈ 𝑒 ↔ 𝑣 ∈ 𝑢))
9	8	3ad2antl2 1162	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑐 = 𝑧 ∧ 𝑓 = 𝑣 ∧ 𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → (𝑓 ∈ 𝑒 ↔ 𝑣 ∈ 𝑢))
10	7, 9	anbi12d 473	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑐 = 𝑧 ∧ 𝑓 = 𝑣 ∧ 𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → ((𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒) ↔ (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
11		simpl3 1004	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑐 = 𝑧 ∧ 𝑓 = 𝑣 ∧ 𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → 𝑏 = 𝑦)
12	10, 11	cbvrexdva2 2726	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 = 𝑧 ∧ 𝑓 = 𝑣 ∧ 𝑏 = 𝑦) → (∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒) ↔ ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
13	5, 12	anbi12d 473	. . . . . . . . . . . . . . . . 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 1211	. . . . . . . . . . . . . . . 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 1205	. . . . . . . . . . . . . . 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 1800	. . . . . . . . . . . . . 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 2046	. . . . . . . . . . . . 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	bitrid 192	. . . . . . . . . . . 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 2475	. . . . . . . . . . . 12 ⊢ (∃!𝑓 ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒) ↔ ∃!𝑓(𝑓 ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒)))
20		df-reu 2475	. . . . . . . . . . . 12 ⊢ (∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∃!𝑣(𝑣 ∈ 𝑧 ∧ ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
21	18, 19, 20	3bitr4g 223	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝑧 ∧ 𝑏 = 𝑦) → (∃!𝑓 ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒) ↔ ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
22	21	adantr 276	. . . . . . . . . 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 527	. . . . . . . . . 10 ⊢ (((𝑐 = 𝑧 ∧ 𝑏 = 𝑦) ∧ 𝑑 = 𝑤) → 𝑐 = 𝑧)
24	22, 23	cbvraldva2 2725	. . . . . . . . 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 268	. . . . . . . 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 476	. . . . . . 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 527	. . . . . . 7 ⊢ (((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) ∧ 𝑐 = 𝑧) → 𝑎 = 𝑥)
28	26, 27	cbvraldva2 2725	. . . . . 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 1941	. . . . 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 1929	. . . 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 1464	. . 3 ⊢ ∀𝑎∃𝑏∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑐 ∃!𝑓 ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒)
33	32	spi 1547	. 2 ⊢ ∃𝑏∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑐 ∃!𝑓 ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒)
34	33	acexmidlemv 5889	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∧ w3a 980  ∀wal 1362  ∃wex 1503  [wsb 1773  ∃!weu 2038  ∀wral 2468  ∃wrex 2469  ∃!wreu 2470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-tr 4117  df-iord 4381  df-on 4383  df-suc 4386  df-iota 5193  df-riota 5847

This theorem is referenced by: (None)