Theorem oaord 4165

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 4164	. . 3 |- ((B e. On /\ C e. On) -> (A e. B -> (C +o A) e. (C +o B)))
2	1	3adant1 795	. 2 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> (C +o A) e. (C +o B)))
3		opreq2 3954	. . . . . 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 4164	. . . . . 6 |- ((A e. On /\ C e. On) -> (B e. A -> (C +o B) e. (C +o A)))
6	5	3adant2 796	. . . . 5 |- ((A e. On /\ B e. On /\ C e. On) -> (B e. A -> (C +o B) e. (C +o A)))
7	4, 6	orim12d 563	. . . 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 95	. . 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 775	. . . . . 6 |- ((A e. On /\ B e. On /\ C e. On) <-> ((A e. On /\ B e. On) /\ C e. On))
10		ancom 435	. . . . . 6 |- (((A e. On /\ B e. On) /\ C e. On) <-> (C e. On /\ (A e. On /\ B e. On)))
11		anandi 509	. . . . . 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	3bitr 177	. . . . 5 |- ((A e. On /\ B e. On /\ C e. On) <-> ((C e. On /\ A e. On) /\ (C e. On /\ B e. On)))
13		oacl 4154	. . . . . . 7 |- ((C e. On /\ A e. On) -> (C +o A) e. On)
14		eloni 2948	. . . . . . 7 |- ((C +o A) e. On -> Ord (C +o A))
15	13, 14	syl 10	. . . . . 6 |- ((C e. On /\ A e. On) -> Ord (C +o A))
16		oacl 4154	. . . . . . 7 |- ((C e. On /\ B e. On) -> (C +o B) e. On)
17		eloni 2948	. . . . . . 7 |- ((C +o B) e. On -> Ord (C +o B))
18	16, 17	syl 10	. . . . . 6 |- ((C e. On /\ B e. On) -> Ord (C +o B))
19	15, 18	anim12i 333	. . . . 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 199	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> (Ord (C +o A) /\ Ord (C +o B)))
21		ordtri2 2972	. . . 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 10	. . 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 783	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. On /\ B e. On))
24		eloni 2948	. . . . 5 |- (A e. On -> Ord A)
25		eloni 2948	. . . . 5 |- (B e. On -> Ord B)
26	24, 25	anim12i 333	. . . 4 |- ((A e. On /\ B e. On) -> (Ord A /\ Ord B))
27		ordtri2 2972	. . . 4 |- ((Ord A /\ Ord B) -> (A e. B <-> -. (A = B \/ B e. A)))
28	23, 26, 27	3syl 20	. . 3 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B <-> -. (A = B \/ B e. A)))
29	8, 22, 28	3imtr4d 541	. 2 |- ((A e. On /\ B e. On /\ C e. On) -> ((C +o A) e. (C +o B) -> A e. B))
30	2, 29	impbid 514	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 146   \/ wo 222   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  Ord word 2937  Oncon0 2938  (class class class)co 3948   +o coa 4114

This theorem is referenced by:  oacan 4166  oaword 4167  oaord1 4169  oa00 4177  oalimcl 4178  oaass 4179  odi 4194  oneo 4196  nnaord 4219

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-9 962  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 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

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 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-oadd 4119

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