Theorem oaord 4181

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 4180	. . 3 |- ((B e. On /\ C e. On) -> (A e. B -> (C +o A) e. (C +o B)))
2	1	3adant1 797	. 2 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> (C +o A) e. (C +o B)))
3		opreq2 3969	. . . . . 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 4180	. . . . . 6 |- ((A e. On /\ C e. On) -> (B e. A -> (C +o B) e. (C +o A)))
6	5	3adant2 798	. . . . 5 |- ((A e. On /\ B e. On /\ C e. On) -> (B e. A -> (C +o B) e. (C +o A)))
7	4, 6	orim12d 565	. . . 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 777	. . . . . 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 510	. . . . . 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 4170	. . . . . . 7 |- ((C e. On /\ A e. On) -> (C +o A) e. On)
14		eloni 2958	. . . . . . 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 4170	. . . . . . 7 |- ((C e. On /\ B e. On) -> (C +o B) e. On)
17		eloni 2958	. . . . . . 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 2982	. . . 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 785	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. On /\ B e. On))
24		eloni 2958	. . . . 5 |- (A e. On -> Ord A)
25		eloni 2958	. . . . 5 |- (B e. On -> Ord B)
26	24, 25	anim12i 333	. . . 4 |- ((A e. On /\ B e. On) -> (Ord A /\ Ord B))
27		ordtri2 2982	. . . 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 543	. 2 |- ((A e. On /\ B e. On /\ C e. On) -> ((C +o A) e. (C +o B) -> A e. B))
30	2, 29	impbid 516	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 775   = wceq 956   e. wcel 958  Ord word 2947  Oncon0 2948  (class class class)co 3963   +o coa 4130

This theorem is referenced by:  oacan 4182  oaword 4183  oaord1 4185  oa00 4193  oalimcl 4194  oaass 4195  odi 4210  oneo 4212  nnaord 4235

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-oadd 4135

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