Theorem ordtri3or 2975

Description: A trichotomy law for ordinals. Proposition 7.10 of [TakeutiZaring] p. 38.

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 2973	. . . . . 6 |- ((Ord A /\ Ord B) -> Ord (A i^i B))
2		ordirr 2962	. . . . . 6 |- (Ord (A i^i B) -> -. (A i^i B) e. (A i^i B))
3	1, 2	syl 10	. . . . 5 |- ((Ord A /\ Ord B) -> -. (A i^i B) e. (A i^i B))
4		elin 2201	. . . . . . . 8 |- ((A i^i B) e. (A i^i B) <-> ((A i^i B) e. A /\ (A i^i B) e. B))
5		incom 2202	. . . . . . . . . 10 |- (A i^i B) = (B i^i A)
6	5	eleq1i 1533	. . . . . . . . 9 |- ((A i^i B) e. B <-> (B i^i A) e. B)
7	6	anbi2i 479	. . . . . . . 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	bitr 173	. . . . . . 7 |- ((A i^i B) e. (A i^i B) <-> ((A i^i B) e. A /\ (B i^i A) e. B))
9	8	negbii 187	. . . . . 6 |- (-. (A i^i B) e. (A i^i B) <-> -. ((A i^i B) e. A /\ (B i^i A) e. B))
10		ianor 305	. . . . . 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	bitr 173	. . . . 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 198	. . . 4 |- ((Ord A /\ Ord B) -> (-. (A i^i B) e. A \/ -. (B i^i A) e. B))
13		inss1 2224	. . . . . . . . . 10 |- (A i^i B) (_ A
14		ordsseleq 2972	. . . . . . . . . 10 |- ((Ord (A i^i B) /\ Ord A) -> ((A i^i B) (_ A <-> ((A i^i B) e. A \/ (A i^i B) = A)))
15	13, 14	mpbii 193	. . . . . . . . 9 |- ((Ord (A i^i B) /\ Ord A) -> ((A i^i B) e. A \/ (A i^i B) = A))
16	15, 1	sylan 448	. . . . . . . 8 |- (((Ord A /\ Ord B) /\ Ord A) -> ((A i^i B) e. A \/ (A i^i B) = A))
17	16	anabss1 498	. . . . . . 7 |- ((Ord A /\ Ord B) -> ((A i^i B) e. A \/ (A i^i B) = A))
18	17	ord 232	. . . . . 6 |- ((Ord A /\ Ord B) -> (-. (A i^i B) e. A -> (A i^i B) = A))
19		df-ss 2047	. . . . . 6 |- (A (_ B <-> (A i^i B) = A)
20	18, 19	syl6ibr 213	. . . . 5 |- ((Ord A /\ Ord B) -> (-. (A i^i B) e. A -> A (_ B))
21		inss1 2224	. . . . . . . . . 10 |- (B i^i A) (_ B
22		ordsseleq 2972	. . . . . . . . . 10 |- ((Ord (B i^i A) /\ Ord B) -> ((B i^i A) (_ B <-> ((B i^i A) e. B \/ (B i^i A) = B)))
23	21, 22	mpbii 193	. . . . . . . . 9 |- ((Ord (B i^i A) /\ Ord B) -> ((B i^i A) e. B \/ (B i^i A) = B))
24		ordin 2973	. . . . . . . . 9 |- ((Ord B /\ Ord A) -> Ord (B i^i A))
25	23, 24	sylan 448	. . . . . . . 8 |- (((Ord B /\ Ord A) /\ Ord B) -> ((B i^i A) e. B \/ (B i^i A) = B))
26	25	anabss4 500	. . . . . . 7 |- ((Ord A /\ Ord B) -> ((B i^i A) e. B \/ (B i^i A) = B))
27	26	ord 232	. . . . . 6 |- ((Ord A /\ Ord B) -> (-. (B i^i A) e. B -> (B i^i A) = B))
28		df-ss 2047	. . . . . 6 |- (B (_ A <-> (B i^i A) = B)
29	27, 28	syl6ibr 213	. . . . 5 |- ((Ord A /\ Ord B) -> (-. (B i^i A) e. B -> B (_ A))
30	20, 29	orim12d 563	. . . 4 |- ((Ord A /\ Ord B) -> ((-. (A i^i B) e. A \/ -. (B i^i A) e. B) -> (A (_ B \/ B (_ A)))
31	12, 30	mpd 26	. . 3 |- ((Ord A /\ Ord B) -> (A (_ B \/ B (_ A))
32		ordsseleq 2972	. . . 4 |- ((Ord A /\ Ord B) -> (A (_ B <-> (A e. B \/ A = B)))
33		ordsseleq 2972	. . . . 5 |- ((Ord B /\ Ord A) -> (B (_ A <-> (B e. A \/ B = A)))
34	33	ancoms 436	. . . 4 |- ((Ord A /\ Ord B) -> (B (_ A <-> (B e. A \/ B = A)))
35	32, 34	orbi12d 625	. . 3 |- ((Ord A /\ Ord B) -> ((A (_ B \/ B (_ A) <-> ((A e. B \/ A = B) \/ (B e. A \/ B = A))))
36	31, 35	mpbid 195	. 2 |- ((Ord A /\ Ord B) -> ((A e. B \/ A = B) \/ (B e. A \/ B = A)))
37		df-3or 774	. . . 4 |- ((A e. B \/ A = B \/ B e. A) <-> ((A e. B \/ A = B) \/ B e. A))
38		or23 263	. . . 4 |- (((A e. B \/ A = B) \/ B e. A) <-> ((A e. B \/ B e. A) \/ A = B))
39	37, 38	bitr 173	. . 3 |- ((A e. B \/ A = B \/ B e. A) <-> ((A e. B \/ B e. A) \/ A = B))
40		orordir 267	. . 3 |- (((A e. B \/ B e. A) \/ A = B) <-> ((A e. B \/ A = B) \/ (B e. A \/ A = B)))
41		eqcom 1473	. . . . 5 |- (A = B <-> B = A)
42	41	orbi2i 255	. . . 4 |- ((B e. A \/ A = B) <-> (B e. A \/ B = A))
43	42	orbi2i 255	. . 3 |- (((A e. B \/ A = B) \/ (B e. A \/ A = B)) <-> ((A e. B \/ A = B) \/ (B e. A \/ B = A)))
44	39, 40, 43	3bitr 177	. 2 |- ((A e. B \/ A = B \/ B e. A) <-> ((A e. B \/ A = B) \/ (B e. A \/ B = A)))
45	36, 44	sylibr 200	1 |- ((Ord A /\ Ord B) -> (A e. B \/ A = B \/ B e. A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   \/ w3o 772   = wceq 953   e. wcel 955   i^i cin 2040   (_ wss 2041  Ord word 2943

This theorem is referenced by:  ordtri1 2977  ordtri3 2980  ordon 2984  ordeleqon 2987  ordtri2or 3074  zorn2lem6 4776

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1120  ax-10o 1137  ax-16 1207  ax-11o 1215  ax-ext 1456  ax-sep 2697  ax-pow 2736  ax-pr 2773

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1169  df-eu 1379  df-mo 1380  df-clab 1461  df-cleq 1466  df-clel 1469  df-ne 1583  df-ral 1645  df-rex 1646  df-v 1807  df-dif 2043  df-un 2044  df-in 2045  df-ss 2047  df-nul 2275  df-pw 2396  df-sn 2406  df-pr 2407  df-op 2410  df-uni 2498  df-br 2614  df-opab 2661  df-tr 2675  df-eprel 2827  df-po 2837  df-so 2847  df-fr 2913  df-we 2930  df-ord 2947

Copyright terms: Public domain