Theorem oeword 4207

Description: Weak ordering property of ordinal exponentiation.

Assertion

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

Proof of Theorem oeword

Step	Hyp	Ref	Expression
1		oeord 4205	. . 3 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> (A e. B <-> (C ^o A) e. (C ^o B)))
2		oecan 4206	. . . . 5 |- (((C e. On /\ A e. On /\ B e. On) /\ 1o e. C) -> ((C ^o A) = (C ^o B) <-> A = B))
3		3anrot 779	. . . . 5 |- ((C e. On /\ A e. On /\ B e. On) <-> (A e. On /\ B e. On /\ C e. On))
4	2, 3	sylanbr 450	. . . 4 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> ((C ^o A) = (C ^o B) <-> A = B))
5	4	bicomd 520	. . 3 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> (A = B <-> (C ^o A) = (C ^o B)))
6	1, 5	orbi12d 626	. 2 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> ((A e. B \/ A = B) <-> ((C ^o A) e. (C ^o B) \/ (C ^o A) = (C ^o B))))
7		onsseleq 2994	. . . 4 |- ((A e. On /\ B e. On) -> (A (_ B <-> (A e. B \/ A = B)))
8	7	3adant3 798	. . 3 |- ((A e. On /\ B e. On /\ C e. On) -> (A (_ B <-> (A e. B \/ A = B)))
9	8	adantr 389	. 2 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> (A (_ B <-> (A e. B \/ A = B)))
10		oecl 4162	. . . . . . . 8 |- ((C e. On /\ A e. On) -> (C ^o A) e. On)
11		oecl 4162	. . . . . . . 8 |- ((C e. On /\ B e. On) -> (C ^o B) e. On)
12	10, 11	anim12i 333	. . . . . . 7 |- (((C e. On /\ A e. On) /\ (C e. On /\ B e. On)) -> ((C ^o A) e. On /\ (C ^o B) e. On))
13	12	anandis 512	. . . . . 6 |- ((C e. On /\ (A e. On /\ B e. On)) -> ((C ^o A) e. On /\ (C ^o B) e. On))
14	13	ancoms 436	. . . . 5 |- (((A e. On /\ B e. On) /\ C e. On) -> ((C ^o A) e. On /\ (C ^o B) e. On))
15	14	3impa 827	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> ((C ^o A) e. On /\ (C ^o B) e. On))
16		onsseleq 2994	. . . 4 |- (((C ^o A) e. On /\ (C ^o B) e. On) -> ((C ^o A) (_ (C ^o B) <-> ((C ^o A) e. (C ^o B) \/ (C ^o A) = (C ^o B))))
17	15, 16	syl 10	. . 3 |- ((A e. On /\ B e. On /\ C e. On) -> ((C ^o A) (_ (C ^o B) <-> ((C ^o A) e. (C ^o B) \/ (C ^o A) = (C ^o B))))
18	17	adantr 389	. 2 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> ((C ^o A) (_ (C ^o B) <-> ((C ^o A) e. (C ^o B) \/ (C ^o A) = (C ^o B))))
19	6, 9, 18	3bitr4d 549	1 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. C) -> (A (_ B <-> (C ^o A) (_ (C ^o B)))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956   (_ wss 2043  Oncon0 2943  (class class class)co 3954  1oc1o 4118   ^o coe 4122

This theorem is referenced by:  oewordi 4208

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1o 4123  df-oadd 4125  df-omul 4126  df-oexp 4127

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