Theorem acexmid 5431

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	|- E.yA.z e. x A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u)

Assertion

Ref	Expression
acexmid	|- (ph \/ -. ph)

Distinct variable group:   x,y,z,w,v,u

Allowed substitution hints:   ph(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 1398	. . . . . . . . . . . . . 14 |- F/v(f e. c /\ E. e e. b ( c e. e /\ f e. e))
2	1	sb8eu 1891	. . . . . . . . . . . . 13 |- (E!f(f e. c /\ E. e e. b ( c e. e /\ f e. e)) <-> E!v[v / f](f e. c /\ E. e e. b ( c e. e /\ f e. e)))
3		eleq12 2080	. . . . . . . . . . . . . . . . . . . 20 |- ((f = v /\ c = z) -> (f e. c <-> v e. z))
4	3	ancoms 255	. . . . . . . . . . . . . . . . . . 19 |- (( c = z /\ f = v) -> (f e. c <-> v e. z))
5	4	3adant3 910	. . . . . . . . . . . . . . . . . 18 |- (( c = z /\ f = v /\ b = y) -> (f e. c <-> v e. z))
6		eleq12 2080	. . . . . . . . . . . . . . . . . . . . 21 |- (( c = z /\ e = u) -> ( c e. e <-> z e. u))
7	6	3ad2antl1 1052	. . . . . . . . . . . . . . . . . . . 20 |- ((( c = z /\ f = v /\ b = y) /\ e = u) -> ( c e. e <-> z e. u))
8		eleq12 2080	. . . . . . . . . . . . . . . . . . . . 21 |- ((f = v /\ e = u) -> (f e. e <-> v e. u))
9	8	3ad2antl2 1053	. . . . . . . . . . . . . . . . . . . 20 |- ((( c = z /\ f = v /\ b = y) /\ e = u) -> (f e. e <-> v e. u))
10	7, 9	anbi12d 445	. . . . . . . . . . . . . . . . . . 19 |- ((( c = z /\ f = v /\ b = y) /\ e = u) -> (( c e. e /\ f e. e) <-> (z e. u /\ v e. u)))
11		simpl3 895	. . . . . . . . . . . . . . . . . . 19 |- ((( c = z /\ f = v /\ b = y) /\ e = u) -> b = y)
12	10, 11	cbvrexdva2 2512	. . . . . . . . . . . . . . . . . 18 |- (( c = z /\ f = v /\ b = y) -> (E. e e. b ( c e. e /\ f e. e) <-> E.u e. y (z e. u /\ v e. u)))
13	5, 12	anbi12d 445	. . . . . . . . . . . . . . . . 17 |- (( c = z /\ f = v /\ b = y) -> ((f e. c /\ E. e e. b ( c e. e /\ f e. e)) <-> (v e. z /\ E.u e. y (z e. u /\ v e. u))))
14	13	3com23 1094	. . . . . . . . . . . . . . . 16 |- (( c = z /\ b = y /\ f = v) -> ((f e. c /\ E. e e. b ( c e. e /\ f e. e)) <-> (v e. z /\ E.u e. y (z e. u /\ v e. u))))
15	14	3expa 1088	. . . . . . . . . . . . . . 15 |- ((( c = z /\ b = y) /\ f = v) -> ((f e. c /\ E. e e. b ( c e. e /\ f e. e)) <-> (v e. z /\ E.u e. y (z e. u /\ v e. u))))
16	15	sbiedv 1650	. . . . . . . . . . . . . 14 |- (( c = z /\ b = y) -> ([v / f](f e. c /\ E. e e. b ( c e. e /\ f e. e)) <-> (v e. z /\ E.u e. y (z e. u /\ v e. u))))
17	16	eubidv 1886	. . . . . . . . . . . . 13 |- (( c = z /\ b = y) -> (E!v[v / f](f e. c /\ E. e e. b ( c e. e /\ f e. e)) <-> E!v(v e. z /\ E.u e. y (z e. u /\ v e. u))))
18	2, 17	syl5bb 181	. . . . . . . . . . . 12 |- (( c = z /\ b = y) -> (E!f(f e. c /\ E. e e. b ( c e. e /\ f e. e)) <-> E!v(v e. z /\ E.u e. y (z e. u /\ v e. u))))
19		df-reu 2287	. . . . . . . . . . . 12 |- (E!f e. c E. e e. b ( c e. e /\ f e. e) <-> E!f(f e. c /\ E. e e. b ( c e. e /\ f e. e)))
20		df-reu 2287	. . . . . . . . . . . 12 |- (E!v e. z E.u e. y (z e. u /\ v e. u) <-> E!v(v e. z /\ E.u e. y (z e. u /\ v e. u)))
21	18, 19, 20	3bitr4g 212	. . . . . . . . . . 11 |- (( c = z /\ b = y) -> (E!f e. c E. e e. b ( c e. e /\ f e. e) <-> E!v e. z E.u e. y (z e. u /\ v e. u)))
22	21	adantr 261	. . . . . . . . . 10 |- ((( c = z /\ b = y) /\ d = w) -> (E!f e. c E. e e. b ( c e. e /\ f e. e) <-> E!v e. z E.u e. y (z e. u /\ v e. u)))
23		simpll 469	. . . . . . . . . 10 |- ((( c = z /\ b = y) /\ d = w) -> c = z)
24	22, 23	cbvraldva2 2511	. . . . . . . . 9 |- (( c = z /\ b = y) -> (A. d e. c E!f e. c E. e e. b ( c e. e /\ f e. e) <-> A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u)))
25	24	ancoms 255	. . . . . . . 8 |- (( b = y /\ c = z) -> (A. d e. c E!f e. c E. e e. b ( c e. e /\ f e. e) <-> A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u)))
26	25	adantll 448	. . . . . . 7 |- ((( a = x /\ b = y) /\ c = z) -> (A. d e. c E!f e. c E. e e. b ( c e. e /\ f e. e) <-> A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u)))
27		simpll 469	. . . . . . 7 |- ((( a = x /\ b = y) /\ c = z) -> a = x)
28	26, 27	cbvraldva2 2511	. . . . . 6 |- (( a = x /\ b = y) -> (A. c e. a A. d e. c E!f e. c E. e e. b ( c e. e /\ f e. e) <-> A.z e. x A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u)))
29	28	cbvexdva 1782	. . . . 5 |- ( a = x -> (E. bA. c e. a A. d e. c E!f e. c E. e e. b ( c e. e /\ f e. e) <-> E.yA.z e. x A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u)))
30	29	cbvalv 1772	. . . 4 |- (A. aE. bA. c e. a A. d e. c E!f e. c E. e e. b ( c e. e /\ f e. e) <-> A.xE.yA.z e. x A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u))
31		acexmid.choice	. . . 4 |- E.yA.z e. x A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u)
32	30, 31	mpgbir 1318	. . 3 |- A. aE. bA. c e. a A. d e. c E!f e. c E. e e. b ( c e. e /\ f e. e)
33	32	spi 1407	. 2 |- E. bA. c e. a A. d e. c E!f e. c E. e e. b ( c e. e /\ f e. e)
34	33	acexmidlemv 5430	1 |- (ph \/ -. ph)

Syntax hints:  -. wn 3   /\ wa 97   <-> wb 98   \/ wo 616   /\ w3a 871  A.wal 1224  E.wex 1358  [wsb 1623  E!weu 1878  A.wral 2280  E.wrex 2281  E!wreu 2282

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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914

This theorem depends on definitions:  df-bi 110  df-3or 872  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-uni 3551  df-tr 3825  df-iord 4048  df-on 4050  df-suc 4053  df-iota 4790  df-riota 5389

This theorem is referenced by: (None)