Theorem ordtri3or 3691

Description: A trichotomy law for ordinals. Proposition 7.10 of [TakeutiZaring] p. 38. (The proof was shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ordtri3or	|- ((Ord A /\ Ord B) -> (A e. B \/ A = B \/ B e. A))

Proof of Theorem ordtri3or

Step	Hyp	Ref	Expression
1		ordin 3689	. . . . . 6 |- ((Ord A /\ Ord B) -> Ord (A i^i B))
2		ordirr 3676	. . . . . 6 |- (Ord (A i^i B) -> -. (A i^i B) e. (A i^i B))
3	1, 2	syl 12	. . . . 5 |- ((Ord A /\ Ord B) -> -. (A i^i B) e. (A i^i B))
4		elin 2786	. . . . . . . 8 |- ((A i^i B) e. (A i^i B) <-> ((A i^i B) e. A /\ (A i^i B) e. B))
5		incom 2787	. . . . . . . . . 10 |- (A i^i B) = (B i^i A)
6	5	eleq1i 1960	. . . . . . . . 9 |- ((A i^i B) e. B <-> (B i^i A) e. B)
7	6	anbi2i 538	. . . . . . . 8 |- (((A i^i B) e. A /\ (A i^i B) e. B) <-> ((A i^i B) e. A /\ (B i^i A) e. B))
8	4, 7	bitri 190	. . . . . . 7 |- ((A i^i B) e. (A i^i B) <-> ((A i^i B) e. A /\ (B i^i A) e. B))
9	8	notbii 204	. . . . . 6 |- (-. (A i^i B) e. (A i^i B) <-> -. ((A i^i B) e. A /\ (B i^i A) e. B))
10		ianor 329	. . . . . 6 |- (-. ((A i^i B) e. A /\ (B i^i A) e. B) <-> (-. (A i^i B) e. A \/ -. (B i^i A) e. B))
11	9, 10	bitri 190	. . . . 5 |- (-. (A i^i B) e. (A i^i B) <-> (-. (A i^i B) e. A \/ -. (B i^i A) e. B))
12	3, 11	sylib 215	. . . 4 |- ((Ord A /\ Ord B) -> (-. (A i^i B) e. A \/ -. (B i^i A) e. B))
13		inss1 2812	. . . . . . . . . 10 |- (A i^i B) C_ A
14		ordsseleq 3687	. . . . . . . . . 10 |- ((Ord (A i^i B) /\ Ord A) -> ((A i^i B) C_ A <-> ((A i^i B) e. A \/ (A i^i B) = A)))
15	13, 14	mpbii 210	. . . . . . . . 9 |- ((Ord (A i^i B) /\ Ord A) -> ((A i^i B) e. A \/ (A i^i B) = A))
16	15, 1	sylan 497	. . . . . . . 8 |- (((Ord A /\ Ord B) /\ Ord A) -> ((A i^i B) e. A \/ (A i^i B) = A))
17	16	anabss1 557	. . . . . . 7 |- ((Ord A /\ Ord B) -> ((A i^i B) e. A \/ (A i^i B) = A))
18	17	ord 249	. . . . . 6 |- ((Ord A /\ Ord B) -> (-. (A i^i B) e. A -> (A i^i B) = A))
19		df-ss 2605	. . . . . 6 |- (A C_ B <-> (A i^i B) = A)
20	18, 19	syl6ibr 230	. . . . 5 |- ((Ord A /\ Ord B) -> (-. (A i^i B) e. A -> A C_ B))
21		inss1 2812	. . . . . . . . . 10 |- (B i^i A) C_ B
22		ordsseleq 3687	. . . . . . . . . 10 |- ((Ord (B i^i A) /\ Ord B) -> ((B i^i A) C_ B <-> ((B i^i A) e. B \/ (B i^i A) = B)))
23	21, 22	mpbii 210	. . . . . . . . 9 |- ((Ord (B i^i A) /\ Ord B) -> ((B i^i A) e. B \/ (B i^i A) = B))
24		ordin 3689	. . . . . . . . 9 |- ((Ord B /\ Ord A) -> Ord (B i^i A))
25	23, 24	sylan 497	. . . . . . . 8 |- (((Ord B /\ Ord A) /\ Ord B) -> ((B i^i A) e. B \/ (B i^i A) = B))
26	25	anabss4 559	. . . . . . 7 |- ((Ord A /\ Ord B) -> ((B i^i A) e. B \/ (B i^i A) = B))
27	26	ord 249	. . . . . 6 |- ((Ord A /\ Ord B) -> (-. (B i^i A) e. B -> (B i^i A) = B))
28		df-ss 2605	. . . . . 6 |- (B C_ A <-> (B i^i A) = B)
29	27, 28	syl6ibr 230	. . . . 5 |- ((Ord A /\ Ord B) -> (-. (B i^i A) e. B -> B C_ A))
30	20, 29	orim12d 624	. . . 4 |- ((Ord A /\ Ord B) -> ((-. (A i^i B) e. A \/ -. (B i^i A) e. B) -> (A C_ B \/ B C_ A)))
31	12, 30	mpd 29	. . 3 |- ((Ord A /\ Ord B) -> (A C_ B \/ B C_ A))
32		sspsstri 2711	. . 3 |- ((A C_ B \/ B C_ A) <-> (A C. B \/ A = B \/ B C. A))
33	31, 32	sylib 215	. 2 |- ((Ord A /\ Ord B) -> (A C. B \/ A = B \/ B C. A))
34		ordelpss 3686	. . 3 |- ((Ord A /\ Ord B) -> (A e. B <-> A C. B))
35		biidd 188	. . 3 |- ((Ord A /\ Ord B) -> (A = B <-> A = B))
36		ordelpss 3686	. . . 4 |- ((Ord B /\ Ord A) -> (B e. A <-> B C. A))
37	36	ancoms 484	. . 3 |- ((Ord A /\ Ord B) -> (B e. A <-> B C. A))
38	34, 35, 37	3orbi123d 1167	. 2 |- ((Ord A /\ Ord B) -> ((A e. B \/ A = B \/ B e. A) <-> (A C. B \/ A = B \/ B C. A)))
39	33, 38	mpbird 213	1 |- ((Ord A /\ Ord B) -> (A e. B \/ A = B \/ B e. A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   \/ w3o 857   = wceq 1298   e. wcel 1300   i^i cin 2592   C_ wss 2593   C. wpss 2594  Ord word 3656

This theorem is referenced by:  ordtri1 3693  ordtri1OLD 3694  ordtri3 3697  ordtri3OLD 3698  ordtri2orOLD 3767  ordon 3863  ordeleqon 3866  zorn2lem6 5955  poseq 13954  soseq 13955  celsor 14443

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660

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