Theorem oeordsuc 5269

Description: Ordering property of ordinal exponentiation with a successor exponent. Corollary 8.36 of [TakeutiZaring] p. 68.

Assertion

Ref	Expression
oeordsuc	|- ((B e. On /\ C e. On) -> (A e. B -> (A ^o suc C) e. (B ^o suc C)))

Proof of Theorem oeordsuc

Step	Hyp	Ref	Expression
1		onelon 3683	. . . 4 |- ((B e. On /\ A e. B) -> A e. On)
2	1	ex 402	. . 3 |- (B e. On -> (A e. B -> A e. On))
3	2	adantr 425	. 2 |- ((B e. On /\ C e. On) -> (A e. B -> A e. On))
4		oewordri 5267	. . . . . . . . . . 11 |- ((B e. On /\ C e. On) -> (A e. B -> (A ^o C) C_ (B ^o C)))
5	4	3adant1 894	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> (A ^o C) C_ (B ^o C)))
6		oecl 5218	. . . . . . . . . . . 12 |- ((A e. On /\ C e. On) -> (A ^o C) e. On)
7	6	3adant2 895	. . . . . . . . . . 11 |- ((A e. On /\ B e. On /\ C e. On) -> (A ^o C) e. On)
8		oecl 5218	. . . . . . . . . . . 12 |- ((B e. On /\ C e. On) -> (B ^o C) e. On)
9	8	3adant1 894	. . . . . . . . . . 11 |- ((A e. On /\ B e. On /\ C e. On) -> (B ^o C) e. On)
10		simp1 876	. . . . . . . . . . 11 |- ((A e. On /\ B e. On /\ C e. On) -> A e. On)
11		omwordri 5251	. . . . . . . . . . 11 |- (((A ^o C) e. On /\ (B ^o C) e. On /\ A e. On) -> ((A ^o C) C_ (B ^o C) -> ((A ^o C) .o A) C_ ((B ^o C) .o A)))
12	7, 9, 10, 11	syl111anc 1100	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ C e. On) -> ((A ^o C) C_ (B ^o C) -> ((A ^o C) .o A) C_ ((B ^o C) .o A)))
13	5, 12	syld 30	. . . . . . . . 9 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> ((A ^o C) .o A) C_ ((B ^o C) .o A)))
14		oesuc 5211	. . . . . . . . . . 11 |- ((A e. On /\ C e. On) -> (A ^o suc C) = ((A ^o C) .o A))
15	14	3adant2 895	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ C e. On) -> (A ^o suc C) = ((A ^o C) .o A))
16	15	sseq1d 2644	. . . . . . . . 9 |- ((A e. On /\ B e. On /\ C e. On) -> ((A ^o suc C) C_ ((B ^o C) .o A) <-> ((A ^o C) .o A) C_ ((B ^o C) .o A)))
17	13, 16	sylibrd 221	. . . . . . . 8 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> (A ^o suc C) C_ ((B ^o C) .o A)))
18		on0eln0 3718	. . . . . . . . . . . . . 14 |- (B e. On -> ((/) e. B <-> B =/= (/)))
19		ne0i 2881	. . . . . . . . . . . . . 14 |- (A e. B -> B =/= (/))
20	18, 19	syl5bir 227	. . . . . . . . . . . . 13 |- (B e. On -> (A e. B -> (/) e. B))
21	20	adantr 425	. . . . . . . . . . . 12 |- ((B e. On /\ C e. On) -> (A e. B -> (/) e. B))
22		oen0 5261	. . . . . . . . . . . . 13 |- (((B e. On /\ C e. On) /\ (/) e. B) -> (/) e. (B ^o C))
23	22	ex 402	. . . . . . . . . . . 12 |- ((B e. On /\ C e. On) -> ((/) e. B -> (/) e. (B ^o C)))
24	21, 23	syld 30	. . . . . . . . . . 11 |- ((B e. On /\ C e. On) -> (A e. B -> (/) e. (B ^o C)))
25		omordi 5245	. . . . . . . . . . . . . 14 |- (((B e. On /\ (B ^o C) e. On) /\ (/) e. (B ^o C)) -> (A e. B -> ((B ^o C) .o A) e. ((B ^o C) .o B)))
26		simpl 346	. . . . . . . . . . . . . . 15 |- ((B e. On /\ C e. On) -> B e. On)
27	26, 8	jca 310	. . . . . . . . . . . . . 14 |- ((B e. On /\ C e. On) -> (B e. On /\ (B ^o C) e. On))
28	25, 27	sylan 497	. . . . . . . . . . . . 13 |- (((B e. On /\ C e. On) /\ (/) e. (B ^o C)) -> (A e. B -> ((B ^o C) .o A) e. ((B ^o C) .o B)))
29	28	ex 402	. . . . . . . . . . . 12 |- ((B e. On /\ C e. On) -> ((/) e. (B ^o C) -> (A e. B -> ((B ^o C) .o A) e. ((B ^o C) .o B))))
30	29	com23 36	. . . . . . . . . . 11 |- ((B e. On /\ C e. On) -> (A e. B -> ((/) e. (B ^o C) -> ((B ^o C) .o A) e. ((B ^o C) .o B))))
31	24, 30	mpdd 57	. . . . . . . . . 10 |- ((B e. On /\ C e. On) -> (A e. B -> ((B ^o C) .o A) e. ((B ^o C) .o B)))
32	31	3adant1 894	. . . . . . . . 9 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> ((B ^o C) .o A) e. ((B ^o C) .o B)))
33		oesuc 5211	. . . . . . . . . . 11 |- ((B e. On /\ C e. On) -> (B ^o suc C) = ((B ^o C) .o B))
34	33	3adant1 894	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ C e. On) -> (B ^o suc C) = ((B ^o C) .o B))
35	34	eleq2d 1964	. . . . . . . . 9 |- ((A e. On /\ B e. On /\ C e. On) -> (((B ^o C) .o A) e. (B ^o suc C) <-> ((B ^o C) .o A) e. ((B ^o C) .o B)))
36	32, 35	sylibrd 221	. . . . . . . 8 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> ((B ^o C) .o A) e. (B ^o suc C)))
37	17, 36	jcad 661	. . . . . . 7 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> ((A ^o suc C) C_ ((B ^o C) .o A) /\ ((B ^o C) .o A) e. (B ^o suc C))))
38	37	3expa 1067	. . . . . 6 |- (((A e. On /\ B e. On) /\ C e. On) -> (A e. B -> ((A ^o suc C) C_ ((B ^o C) .o A) /\ ((B ^o C) .o A) e. (B ^o suc C))))
39		ontr2 3711	. . . . . . . . 9 |- (((A ^o suc C) e. On /\ (B ^o suc C) e. On) -> (((A ^o suc C) C_ ((B ^o C) .o A) /\ ((B ^o C) .o A) e. (B ^o suc C)) -> (A ^o suc C) e. (B ^o suc C)))
40		oecl 5218	. . . . . . . . 9 |- ((A e. On /\ suc C e. On) -> (A ^o suc C) e. On)
41		oecl 5218	. . . . . . . . 9 |- ((B e. On /\ suc C e. On) -> (B ^o suc C) e. On)
42	39, 40, 41	syl2an 503	. . . . . . . 8 |- (((A e. On /\ suc C e. On) /\ (B e. On /\ suc C e. On)) -> (((A ^o suc C) C_ ((B ^o C) .o A) /\ ((B ^o C) .o A) e. (B ^o suc C)) -> (A ^o suc C) e. (B ^o suc C)))
43	42	anandirs 571	. . . . . . 7 |- (((A e. On /\ B e. On) /\ suc C e. On) -> (((A ^o suc C) C_ ((B ^o C) .o A) /\ ((B ^o C) .o A) e. (B ^o suc C)) -> (A ^o suc C) e. (B ^o suc C)))
44		sucelon 3898	. . . . . . 7 |- (C e. On <-> suc C e. On)
45	43, 44	sylan2b 501	. . . . . 6 |- (((A e. On /\ B e. On) /\ C e. On) -> (((A ^o suc C) C_ ((B ^o C) .o A) /\ ((B ^o C) .o A) e. (B ^o suc C)) -> (A ^o suc C) e. (B ^o suc C)))
46	38, 45	syld 30	. . . . 5 |- (((A e. On /\ B e. On) /\ C e. On) -> (A e. B -> (A ^o suc C) e. (B ^o suc C)))
47	46	exp31 407	. . . 4 |- (A e. On -> (B e. On -> (C e. On -> (A e. B -> (A ^o suc C) e. (B ^o suc C)))))
48	47	com4l 43	. . 3 |- (B e. On -> (C e. On -> (A e. B -> (A e. On -> (A ^o suc C) e. (B ^o suc C)))))
49	48	imp 377	. 2 |- ((B e. On /\ C e. On) -> (A e. B -> (A e. On -> (A ^o suc C) e. (B ^o suc C))))
50	3, 49	mpdd 57	1 |- ((B e. On /\ C e. On) -> (A e. B -> (A ^o suc C) e. (B ^o suc C)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017   C_ wss 2593  (/)c0 2875  Oncon0 3657  suc csuc 3659  (class class class)co 4884   .o comu 5175   ^o coe 5176

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-oexp 5181

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