Theorem ordtriexmid 4619

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 7456 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 3498	. . . 4 |- -. { z e. { (/) } | ph } e. (/)
2		ordtriexmidlem 4617	. . . . . 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 1347	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> ( ( x e. (/) \/ x = (/) \/ (/) e. x ) <-> ( { z e. { (/) } | ph } e. (/) \/ { z e. { (/) } | ph } = (/) \/ (/) e. { z e. { (/) } | ph } ) ) )
7		0elon 4489	. . . . . . . 8 |- (/) e. On
8		0ex 4216	. . . . . . . . 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 1347	. . . . . . . . . 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 2621	. . . . . . . . 9 |- ( ( x e. On /\ y e. On ) -> ( x e. y \/ x = y \/ y e. x ) )
18	8, 15, 17	vtocl 2858	. . . . . . . 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 2870	. . . . . 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 1007	. . . . 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 1469	. . 3 |- ( { z e. { (/) } | ph } = (/) \/ (/) e. { z e. { (/) } | ph } )
25		ordtriexmidlem2 4618	. . . 4 |- ( { z e. { (/) } | ph } = (/) -> -. ph )
26	8	snid 3700	. . . . . 6 |- (/) e. { (/) }
27		biidd 172	. . . . . . 7 |- ( z = (/) -> ( ph <-> ph ) )
28	27	elrab3 2963	. . . . . 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 766	. . 3 |- ( ( { z e. { (/) } | ph } = (/) \/ (/) e. { z e. { (/) } | ph } ) -> ( -. ph \/ ph ) )
32	24, 31	ax-mp 5	. 2 |- ( -. ph \/ ph )
33		orcom 735	. 2 |- ( ( ph \/ -. ph ) <-> ( -. ph \/ ph ) )
34	32, 33	mpbir 146	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   \/ w3o 1003   = wceq 1397   e. wcel 2202   A.wral 2510   {crab 2514   (/)c0 3494   {csn 3669   Oncon0 4460

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465  df-suc 4468

This theorem is referenced by: (None)