Theorem oeord 4199

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 4198	. . 3 |- (((B e. On /\ C e. On) /\ 1o e. C) -> (A e. B -> (C ^o A) e. (C ^o B)))
2	1	3adantl1 801	. 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 3954	. . . . . 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 4198	. . . . . 6 |- (((A e. On /\ C e. On) /\ 1o e. C) -> (B e. A -> (C ^o B) e. (C ^o A)))
6	5	3adantl2 802	. . . . 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 563	. . . 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 2972	. . . . . . . . 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 2948	. . . . . . . . 9 |- ((C ^o A) e. On -> Ord (C ^o A))
11		eloni 2948	. . . . . . . . 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 4156	. . . . . . . 8 |- ((C e. On /\ A e. On) -> (C ^o A) e. On)
14		oecl 4156	. . . . . . . 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 511	. . . . . 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 826	. . . 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 2972	. . . . . 6 |- ((Ord A /\ Ord B) -> (A e. B <-> -. (A = B \/ B e. A)))
21		eloni 2948	. . . . . 6 |- (A e. On -> Ord A)
22		eloni 2948	. . . . . 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 797	. . . 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 541	. 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 514	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 773   = wceq 953   e. wcel 955  Ord word 2937  Oncon0 2938  (class class class)co 3948  1oc1o 4112   ^o coe 4116

This theorem is referenced by:  oeword 4201

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-1o 4117  df-oadd 4119  df-omul 4120  df-oexp 4121

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