Theorem ordtri2orexmid 4621

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 4617	. . . . 5 |- { z e. { (/) } | ph } e. On
3		suc0 4508	. . . . . 6 |- suc (/) = { (/) }
4		0elon 4489	. . . . . . 7 |- (/) e. On
5	4	onsuci 4614	. . . . . 6 |- suc (/) e. On
6	3, 5	eqeltrri 2305	. . . . 5 |- { (/) } e. On
7		eleq1 2294	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> ( x e. y <-> { z e. { (/) } | ph } e. y ) )
8		sseq2 3251	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> ( y C_ x <-> y C_ { z e. { (/) } | ph } ) )
9	7, 8	orbi12d 800	. . . . . 6 |- ( x = { z e. { (/) } | ph } -> ( ( x e. y \/ y C_ x ) <-> ( { z e. { (/) } | ph } e. y \/ y C_ { z e. { (/) } | ph } ) ) )
10		eleq2 2295	. . . . . . 7 |- ( y = { (/) } -> ( { z e. { (/) } | ph } e. y <-> { z e. { (/) } | ph } e. { (/) } ) )
11		sseq1 3250	. . . . . . 7 |- ( y = { (/) } -> ( y C_ { z e. { (/) } | ph } <-> { (/) } C_ { z e. { (/) } | ph } ) )
12	10, 11	orbi12d 800	. . . . . 6 |- ( y = { (/) } -> ( ( { z e. { (/) } | ph } e. y \/ y C_ { z e. { (/) } | ph } ) <-> ( { z e. { (/) } | ph } e. { (/) } \/ { (/) } C_ { z e. { (/) } | ph } ) ) )
13	9, 12	rspc2va 2924	. . . . 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 436	. . . 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 5	. . 3 |- ( { z e. { (/) } | ph } e. { (/) } \/ { (/) } C_ { z e. { (/) } | ph } )
16		elsni 3687	. . . . 5 |- ( { z e. { (/) } | ph } e. { (/) } -> { z e. { (/) } | ph } = (/) )
17		ordtriexmidlem2 4618	. . . . 5 |- ( { z e. { (/) } | ph } = (/) -> -. ph )
18	16, 17	syl 14	. . . 4 |- ( { z e. { (/) } | ph } e. { (/) } -> -. ph )
19		snssg 3807	. . . . . 6 |- ( (/) e. On -> ( (/) e. { z e. { (/) } | ph } <-> { (/) } C_ { z e. { (/) } | ph } ) )
20	4, 19	ax-mp 5	. . . . 5 |- ( (/) e. { z e. { (/) } | ph } <-> { (/) } C_ { z e. { (/) } | ph } )
21		0ex 4216	. . . . . . . 8 |- (/) e. _V
22	21	snid 3700	. . . . . . 7 |- (/) e. { (/) }
23		biidd 172	. . . . . . . 8 |- ( z = (/) -> ( ph <-> ph ) )
24	23	elrab3 2963	. . . . . . 7 |- ( (/) e. { (/) } -> ( (/) e. { z e. { (/) } | ph } <-> ph ) )
25	22, 24	ax-mp 5	. . . . . 6 |- ( (/) e. { z e. { (/) } | ph } <-> ph )
26	25	biimpi 120	. . . . 5 |- ( (/) e. { z e. { (/) } | ph } -> ph )
27	20, 26	sylbir 135	. . . 4 |- ( { (/) } C_ { z e. { (/) } | ph } -> ph )
28	18, 27	orim12i 766	. . 3 |- ( ( { z e. { (/) } | ph } e. { (/) } \/ { (/) } C_ { z e. { (/) } | ph } ) -> ( -. ph \/ ph ) )
29	15, 28	ax-mp 5	. 2 |- ( -. ph \/ ph )
30		orcom 735	. 2 |- ( ( -. ph \/ ph ) <-> ( ph \/ -. ph ) )
31	29, 30	mpbi 145	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   <-> wb 105   \/ wo 715   = wceq 1397   e. wcel 2202   A.wral 2510   {crab 2514   C_ wss 3200   (/)c0 3494   {csn 3669   Oncon0 4460   suc csuc 4462

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-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  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-pr 3676  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465  df-suc 4468

This theorem is referenced by: (None)