Theorem oaord 4179

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 4178	. . 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 3967	. . . . . 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 4178	. . . . . 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 4168	. . . . . . 7 |- ((C e. On /\ A e. On) -> (C +o A) e. On)
14		eloni 2956	. . . . . . 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 4168	. . . . . . 7 |- ((C e. On /\ B e. On) -> (C +o B) e. On)
17		eloni 2956	. . . . . . 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 2980	. . . 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 2956	. . . . 5 |- (A e. On -> Ord A)
25		eloni 2956	. . . . 5 |- (B e. On -> Ord B)
26	24, 25	anim12i 333	. . . 4 |- ((A e. On /\ B e. On) -> (Ord A /\ Ord B))
27		ordtri2 2980	. . . 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 2945  Oncon0 2946  (class class class)co 3961   +o coa 4128

This theorem is referenced by:  oacan 4180  oaword 4181  oaord1 4183  oa00 4191  oalimcl 4192  oaass 4193  odi 4208  oneo 4210  nnaord 4233

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 2691  ax-sep 2701  ax-nul 2708  ax-pow 2740  ax-pr 2777  ax-un 2864

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 1586  df-ral 1648  df-rex 1649  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2502  df-iun 2566  df-br 2618  df-opab 2665  df-tr 2679  df-eprel 2830  df-id 2833  df-po 2838  df-so 2848  df-fr 2915  df-we 2932  df-ord 2949  df-on 2950  df-lim 2951  df-suc 2952  df-xp 3182  df-rel 3183  df-cnv 3184  df-co 3185  df-dm 3186  df-rn 3187  df-res 3188  df-ima 3189  df-fun 3190  df-fn 3191  df-fv 3196  df-rdg 3930  df-opr 3963  df-oprab 3964  df-oadd 4133

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