Theorem oeordsuc 4205

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 2962	. . . 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 4203	. . . . . . . . . . 11 |- ((B e. On /\ C e. On) -> (A e. B -> (A ^o C) (_ (B ^o C)))
5	4	3adant1 795	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ C e. On) -> (A e. B -> (A ^o C) (_ (B ^o C)))
6		omwordri 4187	. . . . . . . . . . 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 4156	. . . . . . . . . . . 12 |- ((A e. On /\ C e. On) -> (A ^o C) e. On)
8	7	3adant2 796	. . . . . . . . . . 11 |- ((A e. On /\ B e. On /\ C e. On) -> (A ^o C) e. On)
9		oecl 4156	. . . . . . . . . . . 12 |- ((B e. On /\ C e. On) -> (B ^o C) e. On)
10	9	3adant1 795	. . . . . . . . . . 11 |- ((A e. On /\ B e. On /\ C e. On) -> (B ^o C) e. On)
11		3simp1 786	. . . . . . . . . . 11 |- ((A e. On /\ B e. On /\ C e. On) -> A e. On)
12	6, 8, 10, 11	syl3anc 856	. . . . . . . . . 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 4150	. . . . . . . . . . 11 |- ((A e. On /\ C e. On) -> (A ^o suc C) = ((A ^o C) .o A))
15	14	3adant2 796	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ C e. On) -> (A ^o suc C) = ((A ^o C) .o A))
16	15	sseq1d 2078	. . . . . . . . 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 3014	. . . . . . . . . . . . . 14 |- (B e. On -> ((/) e. B <-> B =/= (/)))
19		ne0i 2276	. . . . . . . . . . . . . 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 4197	. . . . . . . . . . . . 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 4181	. . . . . . . . . . . . . 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 795	. . . . . . . . 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 4150	. . . . . . . . . . 11 |- ((B e. On /\ C e. On) -> (B ^o suc C) = ((B ^o C) .o B))
34	33	3adant1 795	. . . . . . . . . 10 |- ((A e. On /\ B e. On /\ C e. On) -> (B ^o suc C) = ((B ^o C) .o B))
35	34	eleq2d 1533	. . . . . . . . 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 598	. . . . . . 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 831	. . . . . 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 2994	. . . . . . . . 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 4156	. . . . . . . . 9 |- ((A e. On /\ suc C e. On) -> (A ^o suc C) e. On)
41		oecl 4156	. . . . . . . . 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 512	. . . . . . 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 3058	. . . . . . 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 773   = wceq 953   e. wcel 955   =/= wne 1577   (_ wss 2037  (/)c0 2270  Oncon0 2938  suc csuc 2940  (class class class)co 3948   .o comu 4115   ^o coe 4116

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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