Theorem ordtri2orexmid 4191

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 4188	. . . . 5 |- {z e. { (/) } | ph } e. On
3		suc0 4093	. . . . . 6 |- suc (/) = { (/) }
4		0elon 4074	. . . . . . 7 |- (/) e. On
5	4	onsuci 4187	. . . . . 6 |- suc (/) e. On
6	3, 5	eqeltrri 2089	. . . . 5 |- { (/) } e. On
7		eleq1 2078	. . . . . . 7 |- (x = {z e. { (/) } | ph } -> (x e. y <-> {z e. { (/) } | ph } e. y))
8		sseq2 2940	. . . . . . 7 |- (x = {z e. { (/) } | ph } -> (y C_ x <-> y C_ {z e. { (/) } | ph }))
9	7, 8	orbi12d 694	. . . . . 6 |- (x = {z e. { (/) } | ph } -> ((x e. y \/ y C_ x) <-> ( {z e. { (/) } | ph } e. y \/ y C_ {z e. { (/) } | ph })))
10		eleq2 2079	. . . . . . 7 |- (y = { (/) } -> ( {z e. { (/) } | ph } e. y <-> {z e. { (/) } | ph } e. { (/) }))
11		sseq1 2939	. . . . . . 7 |- (y = { (/) } -> (y C_ {z e. { (/) } | ph } <-> { (/) } C_ {z e. { (/) } | ph }))
12	10, 11	orbi12d 694	. . . . . 6 |- (y = { (/) } -> (( {z e. { (/) } | ph } e. y \/ y C_ {z e. { (/) } | ph }) <-> ( {z e. { (/) } | ph } e. { (/) } \/ { (/) } C_ {z e. { (/) } | ph })))
13	9, 12	rspc2va 2636	. . . . 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 414	. . . 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 3370	. . . . 5 |- ( {z e. { (/) } | ph } e. { (/) } -> {z e. { (/) } | ph } = (/))
17		ordtriexmidlem2 4189	. . . . 5 |- ( {z e. { (/) } | ph } = (/) -> -. ph)
18	16, 17	syl 14	. . . 4 |- ( {z e. { (/) } | ph } e. { (/) } -> -. ph)
19		snssg 3470	. . . . . 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 3854	. . . . . . . 8 |- (/) e. _V
22	21	snid 3373	. . . . . . 7 |- (/) e. { (/) }
23		biidd 161	. . . . . . . 8 |- (z = (/) -> (ph <-> ph))
24	23	elrab3 2672	. . . . . . 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 663	. . 3 |- (( {z e. { (/) } | ph } e. { (/) } \/ { (/) } C_ {z e. { (/) } | ph }) -> (-. ph \/ ph))
29	15, 28	ax-mp 7	. 2 |- (-. ph \/ ph)
30		orcom 634	. 2 |- ((-. ph \/ ph) <-> (ph \/ -. ph))
31	29, 30	mpbi 133	1 |- (ph \/ -. ph)

Syntax hints:  -. wn 3   <-> wb 98   \/ wo 616   = wceq 1226   e. wcel 1370  A.wral 2280   {crab 2284   C_ wss 2890   (/)c0 3197   {csn 3346   Oncon0 4045   suc csuc 4047

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-13 1381  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  ax-un 4116

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  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

This theorem is referenced by: (None)