Theorem ordtriexmid 4643

Description: Ordinal trichotomy implies the law of the excluded middle (that is, decidability of an arbitrary proposition).

This theorem is stated in "Constructive ordinals", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

Also see exmidontri 7549 which is much the same theorem but biconditionalized and using the EXMID notation. (Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.)

Hypothesis

Ref	Expression
ordtriexmid.1	|- A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x )

Assertion

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

Distinct variable groups:   x, y   ph, x

Allowed substitution hint:   ph( y)

Proof of Theorem ordtriexmid

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3512	. . . 4 |- -. { z e. { (/) } | ph } e. (/)
2		ordtriexmidlem 4641	. . . . . 6 |- { z e. { (/) } | ph } e. On
3		eleq1 2295	. . . . . . . 8 |- ( x = { z e. { (/) } | ph } -> ( x e. (/) <-> { z e. { (/) } | ph } e. (/) ) )
4		eqeq1 2239	. . . . . . . 8 |- ( x = { z e. { (/) } | ph } -> ( x = (/) <-> { z e. { (/) } | ph } = (/) ) )
5		eleq2 2296	. . . . . . . 8 |- ( x = { z e. { (/) } | ph } -> ( (/) e. x <-> (/) e. { z e. { (/) } | ph } ) )
6	3, 4, 5	3orbi123d 1348	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> ( ( x e. (/) \/ x = (/) \/ (/) e. x ) <-> ( { z e. { (/) } | ph } e. (/) \/ { z e. { (/) } | ph } = (/) \/ (/) e. { z e. { (/) } | ph } ) ) )
7		0elon 4513	. . . . . . . 8 |- (/) e. On
8		0ex 4237	. . . . . . . . 9 |- (/) e. _V
9		eleq1 2295	. . . . . . . . . . 11 |- ( y = (/) -> ( y e. On <-> (/) e. On ) )
10	9	anbi2d 464	. . . . . . . . . 10 |- ( y = (/) -> ( ( x e. On /\ y e. On ) <-> ( x e. On /\ (/) e. On ) ) )
11		eleq2 2296	. . . . . . . . . . 11 |- ( y = (/) -> ( x e. y <-> x e. (/) ) )
12		eqeq2 2242	. . . . . . . . . . 11 |- ( y = (/) -> ( x = y <-> x = (/) ) )
13		eleq1 2295	. . . . . . . . . . 11 |- ( y = (/) -> ( y e. x <-> (/) e. x ) )
14	11, 12, 13	3orbi123d 1348	. . . . . . . . . 10 |- ( y = (/) -> ( ( x e. y \/ x = y \/ y e. x ) <-> ( x e. (/) \/ x = (/) \/ (/) e. x ) ) )
15	10, 14	imbi12d 234	. . . . . . . . 9 |- ( y = (/) -> ( ( ( x e. On /\ y e. On ) -> ( x e. y \/ x = y \/ y e. x ) ) <-> ( ( x e. On /\ (/) e. On ) -> ( x e. (/) \/ x = (/) \/ (/) e. x ) ) ) )
16		ordtriexmid.1	. . . . . . . . . 10 |- A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x )
17	16	rspec2 2631	. . . . . . . . 9 |- ( ( x e. On /\ y e. On ) -> ( x e. y \/ x = y \/ y e. x ) )
18	8, 15, 17	vtocl 2869	. . . . . . . 8 |- ( ( x e. On /\ (/) e. On ) -> ( x e. (/) \/ x = (/) \/ (/) e. x ) )
19	7, 18	mpan2 425	. . . . . . 7 |- ( x e. On -> ( x e. (/) \/ x = (/) \/ (/) e. x ) )
20	6, 19	vtoclga 2881	. . . . . 6 |- ( { z e. { (/) } | ph } e. On -> ( { z e. { (/) } | ph } e. (/) \/ { z e. { (/) } | ph } = (/) \/ (/) e. { z e. { (/) } | ph } ) )
21	2, 20	ax-mp 5	. . . . 5 |- ( { z e. { (/) } | ph } e. (/) \/ { z e. { (/) } | ph } = (/) \/ (/) e. { z e. { (/) } | ph } )
22		3orass 1008	. . . . 5 |- ( ( { z e. { (/) } | ph } e. (/) \/ { z e. { (/) } | ph } = (/) \/ (/) e. { z e. { (/) } | ph } ) <-> ( { z e. { (/) } | ph } e. (/) \/ ( { z e. { (/) } | ph } = (/) \/ (/) e. { z e. { (/) } | ph } ) ) )
23	21, 22	mpbi 145	. . . 4 |- ( { z e. { (/) } | ph } e. (/) \/ ( { z e. { (/) } | ph } = (/) \/ (/) e. { z e. { (/) } | ph } ) )
24	1, 23	mtpor 1470	. . 3 |- ( { z e. { (/) } | ph } = (/) \/ (/) e. { z e. { (/) } | ph } )
25		ordtriexmidlem2 4642	. . . 4 |- ( { z e. { (/) } | ph } = (/) -> -. ph )
26	8	snid 3720	. . . . . 6 |- (/) e. { (/) }
27		biidd 172	. . . . . . 7 |- ( z = (/) -> ( ph <-> ph ) )
28	27	elrab3 2974	. . . . . 6 |- ( (/) e. { (/) } -> ( (/) e. { z e. { (/) } | ph } <-> ph ) )
29	26, 28	ax-mp 5	. . . . 5 |- ( (/) e. { z e. { (/) } | ph } <-> ph )
30	29	biimpi 120	. . . 4 |- ( (/) e. { z e. { (/) } | ph } -> ph )
31	25, 30	orim12i 767	. . 3 |- ( ( { z e. { (/) } | ph } = (/) \/ (/) e. { z e. { (/) } | ph } ) -> ( -. ph \/ ph ) )
32	24, 31	ax-mp 5	. 2 |- ( -. ph \/ ph )
33		orcom 736	. 2 |- ( ( ph \/ -. ph ) <-> ( -. ph \/ ph ) )
34	32, 33	mpbir 146	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   \/ w3o 1004   = wceq 1398   e. wcel 2203   A.wral 2520   {crab 2524   (/)c0 3508   {csn 3689   Oncon0 4484

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 4228  ax-nul 4236  ax-pow 4287

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-uni 3915  df-tr 4209  df-iord 4487  df-on 4489  df-suc 4492

This theorem is referenced by: (None)