Theorem ordtri2orexmid 4211

Description: Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.)

Hypothesis

Ref	Expression
ordtri2orexmid.1	|- A.x e. On A.y e. On (x e. y \/ y C_ x)

Assertion

Ref	Expression
ordtri2orexmid	|- (ph \/ -. ph)

Distinct variable group:   ph,x,y

Proof of Theorem ordtri2orexmid

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtri2orexmid.1	. . . 4 |- A.x e. On A.y e. On (x e. y \/ y C_ x)
2		ordtriexmidlem 4208	. . . . 5 |- {z e. { (/) } | ph } e. On
3		suc0 4114	. . . . . 6 |- suc (/) = { (/) }
4		0elon 4095	. . . . . . 7 |- (/) e. On
5	4	onsuci 4207	. . . . . 6 |- suc (/) e. On
6	3, 5	eqeltrri 2108	. . . . 5 |- { (/) } e. On
7		eleq1 2097	. . . . . . 7 |- (x = {z e. { (/) } | ph } -> (x e. y <-> {z e. { (/) } | ph } e. y))
8		sseq2 2961	. . . . . . 7 |- (x = {z e. { (/) } | ph } -> (y C_ x <-> y C_ {z e. { (/) } | ph }))
9	7, 8	orbi12d 706	. . . . . 6 |- (x = {z e. { (/) } | ph } -> ((x e. y \/ y C_ x) <-> ( {z e. { (/) } | ph } e. y \/ y C_ {z e. { (/) } | ph })))
10		eleq2 2098	. . . . . . 7 |- (y = { (/) } -> ( {z e. { (/) } | ph } e. y <-> {z e. { (/) } | ph } e. { (/) }))
11		sseq1 2960	. . . . . . 7 |- (y = { (/) } -> (y C_ {z e. { (/) } | ph } <-> { (/) } C_ {z e. { (/) } | ph }))
12	10, 11	orbi12d 706	. . . . . 6 |- (y = { (/) } -> (( {z e. { (/) } | ph } e. y \/ y C_ {z e. { (/) } | ph }) <-> ( {z e. { (/) } | ph } e. { (/) } \/ { (/) } C_ {z e. { (/) } | ph })))
13	9, 12	rspc2va 2657	. . . . 5 |- ((( {z e. { (/) } | ph } e. On /\ { (/) } e. On) /\ A.x e. On A.y e. On (x e. y \/ y C_ x)) -> ( {z e. { (/) } | ph } e. { (/) } \/ { (/) } C_ {z e. { (/) } | ph }))
14	2, 6, 13	mpanl12 412	. . . 4 |- (A.x e. On A.y e. On (x e. y \/ y C_ x) -> ( {z e. { (/) } | ph } e. { (/) } \/ { (/) } C_ {z e. { (/) } | ph }))
15	1, 14	ax-mp 7	. . 3 |- ( {z e. { (/) } | ph } e. { (/) } \/ { (/) } C_ {z e. { (/) } | ph })
16		elsni 3391	. . . . 5 |- ( {z e. { (/) } | ph } e. { (/) } -> {z e. { (/) } | ph } = (/))
17		ordtriexmidlem2 4209	. . . . 5 |- ( {z e. { (/) } | ph } = (/) -> -. ph)
18	16, 17	syl 14	. . . 4 |- ( {z e. { (/) } | ph } e. { (/) } -> -. ph)
19		snssg 3491	. . . . . 6 |- ( (/) e. On -> ( (/) e. {z e. { (/) } | ph } <-> { (/) } C_ {z e. { (/) } | ph }))
20	4, 19	ax-mp 7	. . . . 5 |- ( (/) e. {z e. { (/) } | ph } <-> { (/) } C_ {z e. { (/) } | ph })
21		0ex 3875	. . . . . . . 8 |- (/) e. _V
22	21	snid 3394	. . . . . . 7 |- (/) e. { (/) }
23		biidd 161	. . . . . . . 8 |- (z = (/) -> (ph <-> ph))
24	23	elrab3 2693	. . . . . . 7 |- ( (/) e. { (/) } -> ( (/) e. {z e. { (/) } | ph } <-> ph))
25	22, 24	ax-mp 7	. . . . . 6 |- ( (/) e. {z e. { (/) } | ph } <-> ph)
26	25	biimpi 113	. . . . 5 |- ( (/) e. {z e. { (/) } | ph } -> ph)
27	20, 26	sylbir 125	. . . 4 |- ( { (/) } C_ {z e. { (/) } | ph } -> ph)
28	18, 27	orim12i 675	. . 3 |- (( {z e. { (/) } | ph } e. { (/) } \/ { (/) } C_ {z e. { (/) } | ph }) -> (-. ph \/ ph))
29	15, 28	ax-mp 7	. 2 |- (-. ph \/ ph)
30		orcom 646	. 2 |- ((-. ph \/ ph) <-> (ph \/ -. ph))
31	29, 30	mpbi 133	1 |- (ph \/ -. ph)

Syntax hints:  -. wn 3   <-> wb 98   \/ wo 628   = wceq 1242   e. wcel 1390  A.wral 2300   {crab 2304   C_ wss 2911   (/)c0 3218   {csn 3367   Oncon0 4066   suc csuc 4068

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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074

This theorem is referenced by: (None)