Theorem acexmid 5921

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 7273 and df-exmid 4228 syntaxes, see exmidac 7276. (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 1542	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑣(𝑓 ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒))
2	1	sb8eu 2058	. . . . . . . . . . . . 13 ⊢ (∃!𝑓(𝑓 ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒)) ↔ ∃!𝑣[𝑣 / 𝑓](𝑓 ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒)))
3		eleq12 2261	. . . . . . . . . . . . . . . . . . . 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 2261	. . . . . . . . . . . . . . . . . . . . 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 2261	. . . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . 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 1803	. . . . . . . . . . . . . 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 2053	. . . . . . . . . . . . 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 2482	. . . . . . . . . . . 12 ⊢ (∃!𝑓 ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒) ↔ ∃!𝑓(𝑓 ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒)))
20		df-reu 2482	. . . . . . . . . . . 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 2736	. . . . . . . . 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 2736	. . . . . 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 1944	. . . . 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 1932	. . . 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 1467	. . 3 ⊢ ∀𝑎∃𝑏∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑐 ∃!𝑓 ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒)
33	32	spi 1550	. 2 ⊢ ∃𝑏∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑐 ∃!𝑓 ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒)
34	33	acexmidlemv 5920	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∧ w3a 980  ∀wal 1362  ∃wex 1506  [wsb 1776  ∃!weu 2045  ∀wral 2475  ∃wrex 2476  ∃!wreu 2477

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840  df-tr 4132  df-iord 4401  df-on 4403  df-suc 4406  df-iota 5219  df-riota 5877

This theorem is referenced by: (None)