Theorem oeordsuc 4211

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 2967	. . . 4 |- ((B e. On /\ A e. B) -> A e. On)
2	1	ex 373	. . 3 |- (B e. On -> (A e. B -> A e. On))
3	2	adantr 389	. 2 |- ((B e. On /\ C e. On) -> (A e. B -> A e. On))
4		oewordri 4209	. . . . . . . . . . 11 |- ((B e. On /\ C e. On) -> (A e. B -> (A ^o C) (_ (B ^o C)))
5	4	3adant1 796	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> (A ^o C) (_ (B ^o C)))
6		omwordri 4193	. . . . . . . . . . 11 |- (((A ^o C) e. On /\ (B ^o C) e. On /\ A e. On) -> ((A ^o C) (_ (B ^o C) -> ((A ^o C) .o A) (_ ((B ^o C) .o A)))
7		oecl 4162	. . . . . . . . . . . 12 |- ((A e. On /\ C e. On) -> (A ^o C) e. On)
8	7	3adant2 797	. . . . . . . . . . 11 |- ((A e. On /\ B e. On /\ C e. On) -> (A ^o C) e. On)
9		oecl 4162	. . . . . . . . . . . 12 |- ((B e. On /\ C e. On) -> (B ^o C) e. On)
10	9	3adant1 796	. . . . . . . . . . 11 |- ((A e. On /\ B e. On /\ C e. On) -> (B ^o C) e. On)
11		3simp1 787	. . . . . . . . . . 11 |- ((A e. On /\ B e. On /\ C e. On) -> A e. On)
12	6, 8, 10, 11	syl3anc 857	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ C e. On) -> ((A ^o C) (_ (B ^o C) -> ((A ^o C) .o A) (_ ((B ^o C) .o A)))
13	5, 12	syld 27	. . . . . . . . 9 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> ((A ^o C) .o A) (_ ((B ^o C) .o A)))
14		oesuc 4156	. . . . . . . . . . 11 |- ((A e. On /\ C e. On) -> (A ^o suc C) = ((A ^o C) .o A))
15	14	3adant2 797	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ C e. On) -> (A ^o suc C) = ((A ^o C) .o A))
16	15	sseq1d 2084	. . . . . . . . 9 |- ((A e. On /\ B e. On /\ C e. On) -> ((A ^o suc C) (_ ((B ^o C) .o A) <-> ((A ^o C) .o A) (_ ((B ^o C) .o A)))
17	13, 16	sylibrd 204	. . . . . . . 8 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> (A ^o suc C) (_ ((B ^o C) .o A)))
18		on0eln0 3019	. . . . . . . . . . . . . 14 |- (B e. On -> ((/) e. B <-> B =/= (/)))
19		ne0i 2282	. . . . . . . . . . . . . 14 |- (A e. B -> B =/= (/))
20	18, 19	syl5bir 210	. . . . . . . . . . . . 13 |- (B e. On -> (A e. B -> (/) e. B))
21	20	adantr 389	. . . . . . . . . . . 12 |- ((B e. On /\ C e. On) -> (A e. B -> (/) e. B))
22		oen0 4203	. . . . . . . . . . . . 13 |- (((B e. On /\ C e. On) /\ (/) e. B) -> (/) e. (B ^o C))
23	22	ex 373	. . . . . . . . . . . 12 |- ((B e. On /\ C e. On) -> ((/) e. B -> (/) e. (B ^o C)))
24	21, 23	syld 27	. . . . . . . . . . 11 |- ((B e. On /\ C e. On) -> (A e. B -> (/) e. (B ^o C)))
25		omordi 4187	. . . . . . . . . . . . . 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		pm3.26 319	. . . . . . . . . . . . . . 15 |- ((B e. On /\ C e. On) -> B e. On)
27	26, 9	jca 288	. . . . . . . . . . . . . 14 |- ((B e. On /\ C e. On) -> (B e. On /\ (B ^o C) e. On))
28	25, 27	sylan 448	. . . . . . . . . . . . 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 373	. . . . . . . . . . . 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 32	. . . . . . . . . . 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 46	. . . . . . . . . 10 |- ((B e. On /\ C e. On) -> (A e. B -> ((B ^o C) .o A) e. ((B ^o C) .o B)))
32	31	3adant1 796	. . . . . . . . 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 4156	. . . . . . . . . . 11 |- ((B e. On /\ C e. On) -> (B ^o suc C) = ((B ^o C) .o B))
34	33	3adant1 796	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ C e. On) -> (B ^o suc C) = ((B ^o C) .o B))
35	34	eleq2d 1538	. . . . . . . . 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 204	. . . . . . . 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 599	. . . . . . 7 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> ((A ^o suc C) (_ ((B ^o C) .o A) /\ ((B ^o C) .o A) e. (B ^o suc C))))
38	37	3expa 832	. . . . . 6 |- (((A e. On /\ B e. On) /\ C e. On) -> (A e. B -> ((A ^o suc C) (_ ((B ^o C) .o A) /\ ((B ^o C) .o A) e. (B ^o suc C))))
39		ontr2 2999	. . . . . . . . 9 |- (((A ^o suc C) e. On /\ (B ^o suc C) e. On) -> (((A ^o suc 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 4162	. . . . . . . . 9 |- ((A e. On /\ suc C e. On) -> (A ^o suc C) e. On)
41		oecl 4162	. . . . . . . . 9 |- ((B e. On /\ suc C e. On) -> (B ^o suc C) e. On)
42	39, 40, 41	syl2an 454	. . . . . . . 8 |- (((A e. On /\ suc C e. On) /\ (B e. On /\ suc C e. On)) -> (((A ^o suc 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 513	. . . . . . 7 |- (((A e. On /\ B e. On) /\ suc C e. On) -> (((A ^o suc 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 3063	. . . . . . 7 |- (C e. On <-> suc C e. On)
45	43, 44	sylan2b 452	. . . . . 6 |- (((A e. On /\ B e. On) /\ C e. On) -> (((A ^o suc 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 27	. . . . 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 376	. . . 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 39	. . 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 350	. 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 46	1 |- ((B e. On /\ C e. On) -> (A e. B -> (A ^o suc C) e. (B ^o suc C)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956   =/= wne 1582   (_ wss 2043  (/)c0 2276  Oncon0 2943  suc csuc 2945  (class class class)co 3954   .o comu 4121   ^o coe 4122

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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