Theorem oeord 4212

Description: Ordering property of ordinal exponentiation. Corollary 8.34 of [TakeutiZaring] p. 68 and its converse.

Assertion

Ref	Expression
oeord	|- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> (A e. B <-> (C ^o A) e. (C ^o B)))

Proof of Theorem oeord

Step	Hyp	Ref	Expression
1		oeordi 4211	. . 3 |- (((B e. On /\ C e. On) /\ 1o e. C) -> (A e. B -> (C ^o A) e. (C ^o B)))
2	1	3adantl1 802	. 2 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> (A e. B -> (C ^o A) e. (C ^o B)))
3		opreq2 3966	. . . . . 6 |- (A = B -> (C ^o A) = (C ^o B))
4	3	a1i 8	. . . . 5 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> (A = B -> (C ^o A) = (C ^o B)))
5		oeordi 4211	. . . . . 6 |- (((A e. On /\ C e. On) /\ 1o e. C) -> (B e. A -> (C ^o B) e. (C ^o A)))
6	5	3adantl2 803	. . . . 5 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> (B e. A -> (C ^o B) e. (C ^o A)))
7	4, 6	orim12d 564	. . . 4 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> ((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) /\ 1o e. C) -> (-. ((C ^o A) = (C ^o B) \/ (C ^o B) e. (C ^o A)) -> -. (A = B \/ B e. A)))
9		ordtri2 2979	. . . . . . . . 9 |- ((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))))
10		eloni 2955	. . . . . . . . 9 |- ((C ^o A) e. On -> Ord (C ^o A))
11		eloni 2955	. . . . . . . . 9 |- ((C ^o B) e. On -> Ord (C ^o B))
12	9, 10, 11	syl2an 454	. . . . . . . 8 |- (((C ^o A) e. On /\ (C ^o B) e. On) -> ((C ^o A) e. (C ^o B) <-> -. ((C ^o A) = (C ^o B) \/ (C ^o B) e. (C ^o A))))
13		oecl 4169	. . . . . . . 8 |- ((C e. On /\ A e. On) -> (C ^o A) e. On)
14		oecl 4169	. . . . . . . 8 |- ((C e. On /\ B e. On) -> (C ^o B) e. On)
15	12, 13, 14	syl2an 454	. . . . . . 7 |- (((C e. On /\ A e. On) /\ (C e. On /\ B e. On)) -> ((C ^o A) e. (C ^o B) <-> -. ((C ^o A) = (C ^o B) \/ (C ^o B) e. (C ^o A))))
16	15	anandis 512	. . . . . 6 |- ((C e. On /\ (A e. On /\ B e. On)) -> ((C ^o A) e. (C ^o B) <-> -. ((C ^o A) = (C ^o B) \/ (C ^o B) e. (C ^o A))))
17	16	ancoms 436	. . . . 5 |- (((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))))
18	17	3impa 827	. . . 4 |- ((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))))
19	18	adantr 389	. . 3 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> ((C ^o A) e. (C ^o B) <-> -. ((C ^o A) = (C ^o B) \/ (C ^o B) e. (C ^o A))))
20		ordtri2 2979	. . . . . 6 |- ((Ord A /\ Ord B) -> (A e. B <-> -. (A = B \/ B e. A)))
21		eloni 2955	. . . . . 6 |- (A e. On -> Ord A)
22		eloni 2955	. . . . . 6 |- (B e. On -> Ord B)
23	20, 21, 22	syl2an 454	. . . . 5 |- ((A e. On /\ B e. On) -> (A e. B <-> -. (A = B \/ B e. A)))
24	23	3adant3 798	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B <-> -. (A = B \/ B e. A)))
25	24	adantr 389	. . 3 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> (A e. B <-> -. (A = B \/ B e. A)))
26	8, 19, 25	3imtr4d 542	. 2 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> ((C ^o A) e. (C ^o B) -> A e. B))
27	2, 26	impbid 515	1 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> (A e. B <-> (C ^o A) e. (C ^o B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957  Ord word 2944  Oncon0 2945  (class class class)co 3960  1oc1o 4125   ^o coe 4129

This theorem is referenced by:  oeword 4214

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  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 2501  df-iun 2565  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-id 2832  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949  df-lim 2950  df-suc 2951  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-fv 3195  df-rdg 3929  df-opr 3962  df-oprab 3963  df-1o 4130  df-oadd 4132  df-omul 4133  df-oexp 4134

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