Theorem ordtri3or 2230

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 2228	. . . . . 6 |- ((Ord A /\ Ord B) -> Ord (A i^i B))
2		ordeirr 2217	. . . . . 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 1635	. . . . . . . 8 |- ((A i^i B) e. (A i^i B) <-> ((A i^i B) e. A /\ (A i^i B) e. B))
5		incom 1636	. . . . . . . . . 10 |- (A i^i B) = (B i^i A)
6	5	eleq1i 1152	. . . . . . . . 9 |- ((A i^i B) e. B <-> (B i^i A) e. B)
7	6	anbi2i 367	. . . . . . . 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 151	. . . . . . 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 162	. . . . . 6 |- (-. (A i^i B) e. (A i^i B) <-> -. ((A i^i B) e. A /\ (B i^i A) e. B))
10		ianor 253	. . . . . 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 151	. . . . 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 173	. . . 4 |- ((Ord A /\ Ord B) -> (-. (A i^i B) e. A \/ -. (B i^i A) e. B))
13		inss1 1657	. . . . . . . . . 10 |- (A i^i B) (_ A
14		ordsseleq 2227	. . . . . . . . . 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 168	. . . . . . . . 9 |- ((Ord (A i^i B) /\ Ord A) -> ((A i^i B) e. A \/ (A i^i B) = A))
16	15, 1	sylan 343	. . . . . . . 8 |- (((Ord A /\ Ord B) /\ Ord A) -> ((A i^i B) e. A \/ (A i^i B) = A))
17	16	anabss1 381	. . . . . . 7 |- ((Ord A /\ Ord B) -> ((A i^i B) e. A \/ (A i^i B) = A))
18	17	ord 202	. . . . . 6 |- ((Ord A /\ Ord B) -> (-. (A i^i B) e. A -> (A i^i B) = A))
19		df-ss 1492	. . . . . 6 |- (A (_ B <-> (A i^i B) = A)
20	18, 19	syl6ibr 186	. . . . 5 |- ((Ord A /\ Ord B) -> (-. (A i^i B) e. A -> A (_ B))
21		inss1 1657	. . . . . . . . . 10 |- (B i^i A) (_ B
22		ordsseleq 2227	. . . . . . . . . 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 168	. . . . . . . . 9 |- ((Ord (B i^i A) /\ Ord B) -> ((B i^i A) e. B \/ (B i^i A) = B))
24		ordin 2228	. . . . . . . . 9 |- ((Ord B /\ Ord A) -> Ord (B i^i A))
25	23, 24	sylan 343	. . . . . . . 8 |- (((Ord B /\ Ord A) /\ Ord B) -> ((B i^i A) e. B \/ (B i^i A) = B))
26	25	anabss4 383	. . . . . . 7 |- ((Ord A /\ Ord B) -> ((B i^i A) e. B \/ (B i^i A) = B))
27	26	ord 202	. . . . . 6 |- ((Ord A /\ Ord B) -> (-. (B i^i A) e. B -> (B i^i A) = B))
28		df-ss 1492	. . . . . 6 |- (B (_ A <-> (B i^i A) = B)
29	27, 28	syl6ibr 186	. . . . 5 |- ((Ord A /\ Ord B) -> (-. (B i^i A) e. B -> B (_ A))
30	20, 29	orim12d 436	. . . 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 46	. . 3 |- ((Ord A /\ Ord B) -> (A (_ B \/ B (_ A))
32		ordsseleq 2227	. . . 4 |- ((Ord A /\ Ord B) -> (A (_ B <-> (A e. B \/ A = B)))
33		ordsseleq 2227	. . . . 5 |- ((Ord B /\ Ord A) -> (B (_ A <-> (B e. A \/ B = A)))
34	33	ancoms 334	. . . 4 |- ((Ord A /\ Ord B) -> (B (_ A <-> (B e. A \/ B = A)))
35	32, 34	orbi12d 475	. . 3 |- ((Ord A /\ Ord B) -> ((A (_ B \/ B (_ A) <-> ((A e. B \/ A = B) \/ (B e. A \/ B = A))))
36	31, 35	mpbid 170	. 2 |- ((Ord A /\ Ord B) -> ((A e. B \/ A = B) \/ (B e. A \/ B = A)))
37		df-3or 582	. . 3 |- ((A e. B \/ A = B \/ B e. A) <-> ((A e. B \/ A = B) \/ B e. A))
38		oridm 208	. . . . . 6 |- ((A = B \/ A = B) <-> A = B)
39		cleqcom 1103	. . . . . . 7 |- (A = B <-> B = A)
40	39	orbi2i 214	. . . . . 6 |- ((A = B \/ A = B) <-> (A = B \/ B = A))
41	38, 40	bitr3 153	. . . . 5 |- (A = B <-> (A = B \/ B = A))
42	41	orbi2i 214	. . . 4 |- (((A e. B \/ B e. A) \/ A = B) <-> ((A e. B \/ B e. A) \/ (A = B \/ B = A)))
43		or23 219	. . . 4 |- (((A e. B \/ A = B) \/ B e. A) <-> ((A e. B \/ B e. A) \/ A = B))
44		or4 220	. . . 4 |- (((A e. B \/ A = B) \/ (B e. A \/ B = A)) <-> ((A e. B \/ B e. A) \/ (A = B \/ B = A)))
45	42, 43, 44	3bitr4 158	. . 3 |- (((A e. B \/ A = B) \/ B e. A) <-> ((A e. B \/ A = B) \/ (B e. A \/ B = A)))
46	37, 45	bitr 151	. 2 |- ((A e. B \/ A = B \/ B e. A) <-> ((A e. B \/ A = B) \/ (B e. A \/ B = A)))
47	36, 46	sylibr 175	1 |- ((Ord A /\ Ord B) -> (A e. B \/ A = B \/ B e. A))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196   \/ w3o 580   = wceq 1091   e. wcel 1092   i^i cin 1486   (_ wss 1487  Ord word 2198

This theorem is referenced by:  ordtri1 2231  ordtri3 2234  ordon 2238  ordeleqon 2241  ordtri2or 2328  zornlem6 3608

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202

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