Theorem oaord 5228

Description: Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58 and its converse.

Assertion

Ref	Expression
oaord	|- ((A e. On /\ B e. On /\ C e. On) -> (A e. B <-> (C +o A) e. (C +o B)))

Proof of Theorem oaord

Step	Hyp	Ref	Expression
1		oaordi 5227	. . 3 |- ((B e. On /\ C e. On) -> (A e. B -> (C +o A) e. (C +o B)))
2	1	3adant1 894	. 2 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> (C +o A) e. (C +o B)))
3		opreq2 4890	. . . . . 6 |- (A = B -> (C +o A) = (C +o B))
4	3	a1i 8	. . . . 5 |- ((A e. On /\ B e. On /\ C e. On) -> (A = B -> (C +o A) = (C +o B)))
5		oaordi 5227	. . . . . 6 |- ((A e. On /\ C e. On) -> (B e. A -> (C +o B) e. (C +o A)))
6	5	3adant2 895	. . . . 5 |- ((A e. On /\ B e. On /\ C e. On) -> (B e. A -> (C +o B) e. (C +o A)))
7	4, 6	orim12d 624	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> ((A = B \/ B e. A) -> ((C +o A) = (C +o B) \/ (C +o B) e. (C +o A))))
8	7	con3d 111	. . 3 |- ((A e. On /\ B e. On /\ C e. On) -> (-. ((C +o A) = (C +o B) \/ (C +o B) e. (C +o A)) -> -. (A = B \/ B e. A)))
9		df-3an 860	. . . . . 6 |- ((A e. On /\ B e. On /\ C e. On) <-> ((A e. On /\ B e. On) /\ C e. On))
10		ancom 482	. . . . . 6 |- (((A e. On /\ B e. On) /\ C e. On) <-> (C e. On /\ (A e. On /\ B e. On)))
11		anandi 568	. . . . . 6 |- ((C e. On /\ (A e. On /\ B e. On)) <-> ((C e. On /\ A e. On) /\ (C e. On /\ B e. On)))
12	9, 10, 11	3bitri 194	. . . . 5 |- ((A e. On /\ B e. On /\ C e. On) <-> ((C e. On /\ A e. On) /\ (C e. On /\ B e. On)))
13		oacl 5215	. . . . . . 7 |- ((C e. On /\ A e. On) -> (C +o A) e. On)
14		eloni 3667	. . . . . . 7 |- ((C +o A) e. On -> Ord (C +o A))
15	13, 14	syl 12	. . . . . 6 |- ((C e. On /\ A e. On) -> Ord (C +o A))
16		oacl 5215	. . . . . . 7 |- ((C e. On /\ B e. On) -> (C +o B) e. On)
17		eloni 3667	. . . . . . 7 |- ((C +o B) e. On -> Ord (C +o B))
18	16, 17	syl 12	. . . . . 6 |- ((C e. On /\ B e. On) -> Ord (C +o B))
19	15, 18	anim12i 360	. . . . 5 |- (((C e. On /\ A e. On) /\ (C e. On /\ B e. On)) -> (Ord (C +o A) /\ Ord (C +o B)))
20	12, 19	sylbi 216	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> (Ord (C +o A) /\ Ord (C +o B)))
21		ordtri2 3696	. . . 4 |- ((Ord (C +o A) /\ Ord (C +o B)) -> ((C +o A) e. (C +o B) <-> -. ((C +o A) = (C +o B) \/ (C +o B) e. (C +o A))))
22	20, 21	syl 12	. . 3 |- ((A e. On /\ B e. On /\ C e. On) -> ((C +o A) e. (C +o B) <-> -. ((C +o A) = (C +o B) \/ (C +o B) e. (C +o A))))
23		3simpa 872	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. On /\ B e. On))
24		eloni 3667	. . . . 5 |- (A e. On -> Ord A)
25		eloni 3667	. . . . 5 |- (B e. On -> Ord B)
26	24, 25	anim12i 360	. . . 4 |- ((A e. On /\ B e. On) -> (Ord A /\ Ord B))
27		ordtri2 3696	. . . 4 |- ((Ord A /\ Ord B) -> (A e. B <-> -. (A = B \/ B e. A)))
28	23, 26, 27	3syl 24	. . 3 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B <-> -. (A = B \/ B e. A)))
29	8, 22, 28	3imtr4d 602	. 2 |- ((A e. On /\ B e. On /\ C e. On) -> ((C +o A) e. (C +o B) -> A e. B))
30	2, 29	impbid 574	1 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B <-> (C +o A) e. (C +o B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  Ord word 3656  Oncon0 3657  (class class class)co 4884   +o coa 5174

This theorem is referenced by:  oacan 5229  oaword 5230  oawordOLD 5231  oaord1 5233  oa00 5241  oalimcl 5242  oaass 5243  odi 5258  oneo 5260  nnaord 5290

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-13 1311  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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

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-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-oadd 5179

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