Theorem acexmid 6048

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 7512 and df-exmid 4307 syntaxes, see exmidac 7515. (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 1577	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑣(𝑓 ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒))
2	1	sb8eu 2093	. . . . . . . . . . . . 13 ⊢ (∃!𝑓(𝑓 ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒)) ↔ ∃!𝑣[𝑣 / 𝑓](𝑓 ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒)))
3		eleq12 2297	. . . . . . . . . . . . . . . . . . . 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 1044	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 = 𝑧 ∧ 𝑓 = 𝑣 ∧ 𝑏 = 𝑦) → (𝑓 ∈ 𝑐 ↔ 𝑣 ∈ 𝑧))
6		eleq12 2297	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑐 = 𝑧 ∧ 𝑒 = 𝑢) → (𝑐 ∈ 𝑒 ↔ 𝑧 ∈ 𝑢))
7	6	3ad2antl1 1186	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑐 = 𝑧 ∧ 𝑓 = 𝑣 ∧ 𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → (𝑐 ∈ 𝑒 ↔ 𝑧 ∈ 𝑢))
8		eleq12 2297	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 = 𝑣 ∧ 𝑒 = 𝑢) → (𝑓 ∈ 𝑒 ↔ 𝑣 ∈ 𝑢))
9	8	3ad2antl2 1187	. . . . . . . . . . . . . . . . . . . 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 1029	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑐 = 𝑧 ∧ 𝑓 = 𝑣 ∧ 𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → 𝑏 = 𝑦)
12	10, 11	cbvrexdva2 2785	. . . . . . . . . . . . . . . . . 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 1236	. . . . . . . . . . . . . . . 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 1230	. . . . . . . . . . . . . . 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 1838	. . . . . . . . . . . . . 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 2088	. . . . . . . . . . . . 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 2527	. . . . . . . . . . . 12 ⊢ (∃!𝑓 ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒) ↔ ∃!𝑓(𝑓 ∈ 𝑐 ∧ ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒)))
20		df-reu 2527	. . . . . . . . . . . 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 2784	. . . . . . . . 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 2784	. . . . . 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 1979	. . . . 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 1967	. . . 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 1502	. . 3 ⊢ ∀𝑎∃𝑏∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑐 ∃!𝑓 ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒)
33	32	spi 1585	. 2 ⊢ ∃𝑏∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑐 ∃!𝑓 ∈ 𝑐 ∃𝑒 ∈ 𝑏 (𝑐 ∈ 𝑒 ∧ 𝑓 ∈ 𝑒)
34	33	acexmidlemv 6047	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 716   ∧ w3a 1005  ∀wal 1396  ∃wex 1541  [wsb 1811  ∃!weu 2080  ∀wral 2520  ∃wrex 2521  ∃!wreu 2522

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-uni 3914  df-tr 4208  df-iord 4486  df-on 4488  df-suc 4491  df-iota 5311  df-riota 6002

This theorem is referenced by: (None)