Theorem ordtriexmid 4625

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 7500 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 3500	. . . 4 |- -. { z e. { (/) } | ph } e. (/)
2		ordtriexmidlem 4623	. . . . . 6 |- { z e. { (/) } | ph } e. On
3		eleq1 2294	. . . . . . . 8 |- ( x = { z e. { (/) } | ph } -> ( x e. (/) <-> { z e. { (/) } | ph } e. (/) ) )
4		eqeq1 2238	. . . . . . . 8 |- ( x = { z e. { (/) } | ph } -> ( x = (/) <-> { z e. { (/) } | ph } = (/) ) )
5		eleq2 2295	. . . . . . . 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 4495	. . . . . . . 8 |- (/) e. On
8		0ex 4221	. . . . . . . . 9 |- (/) e. _V
9		eleq1 2294	. . . . . . . . . . 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 2295	. . . . . . . . . . 11 |- ( y = (/) -> ( x e. y <-> x e. (/) ) )
12		eqeq2 2241	. . . . . . . . . . 11 |- ( y = (/) -> ( x = y <-> x = (/) ) )
13		eleq1 2294	. . . . . . . . . . 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 2622	. . . . . . . . 9 |- ( ( x e. On /\ y e. On ) -> ( x e. y \/ x = y \/ y e. x ) )
18	8, 15, 17	vtocl 2859	. . . . . . . 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 2871	. . . . . 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 4624	. . . 4 |- ( { z e. { (/) } | ph } = (/) -> -. ph )
26	8	snid 3704	. . . . . 6 |- (/) e. { (/) }
27		biidd 172	. . . . . . 7 |- ( z = (/) -> ( ph <-> ph ) )
28	27	elrab3 2964	. . . . . 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 2202   A.wral 2511   {crab 2515   (/)c0 3496   {csn 3673   Oncon0 4466

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 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-uni 3899  df-tr 4193  df-iord 4469  df-on 4471  df-suc 4474

This theorem is referenced by: (None)