Theorem oaordi 5227

Description: Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58.

Assertion

Ref	Expression
oaordi	|- ((B e. On /\ C e. On) -> (A e. B -> (C +o A) e. (C +o B)))

Proof of Theorem oaordi

Step	Hyp	Ref	Expression
1		onelon 3683	. . . . 5 |- ((B e. On /\ A e. B) -> A e. On)
2	1	adantll 428	. . . 4 |- (((C e. On /\ B e. On) /\ A e. B) -> A e. On)
3		eloni 3667	. . . . . . . . 9 |- (B e. On -> Ord B)
4		ordsucss 3899	. . . . . . . . 9 |- (Ord B -> (A e. B -> suc A C_ B))
5	3, 4	syl 12	. . . . . . . 8 |- (B e. On -> (A e. B -> suc A C_ B))
6	5	ad2antlr 441	. . . . . . 7 |- (((C e. On /\ B e. On) /\ A e. On) -> (A e. B -> suc A C_ B))
7		opreq2 4890	. . . . . . . . . . . . . 14 |- (x = suc A -> (C +o x) = (C +o suc A))
8	7	sseq2d 2645	. . . . . . . . . . . . 13 |- (x = suc A -> ((C +o suc A) C_ (C +o x) <-> (C +o suc A) C_ (C +o suc A)))
9	8	imbi2d 674	. . . . . . . . . . . 12 |- (x = suc A -> ((C e. On -> (C +o suc A) C_ (C +o x)) <-> (C e. On -> (C +o suc A) C_ (C +o suc A))))
10		opreq2 4890	. . . . . . . . . . . . . 14 |- (x = y -> (C +o x) = (C +o y))
11	10	sseq2d 2645	. . . . . . . . . . . . 13 |- (x = y -> ((C +o suc A) C_ (C +o x) <-> (C +o suc A) C_ (C +o y)))
12	11	imbi2d 674	. . . . . . . . . . . 12 |- (x = y -> ((C e. On -> (C +o suc A) C_ (C +o x)) <-> (C e. On -> (C +o suc A) C_ (C +o y))))
13		opreq2 4890	. . . . . . . . . . . . . 14 |- (x = suc y -> (C +o x) = (C +o suc y))
14	13	sseq2d 2645	. . . . . . . . . . . . 13 |- (x = suc y -> ((C +o suc A) C_ (C +o x) <-> (C +o suc A) C_ (C +o suc y)))
15	14	imbi2d 674	. . . . . . . . . . . 12 |- (x = suc y -> ((C e. On -> (C +o suc A) C_ (C +o x)) <-> (C e. On -> (C +o suc A) C_ (C +o suc y))))
16		opreq2 4890	. . . . . . . . . . . . . 14 |- (x = B -> (C +o x) = (C +o B))
17	16	sseq2d 2645	. . . . . . . . . . . . 13 |- (x = B -> ((C +o suc A) C_ (C +o x) <-> (C +o suc A) C_ (C +o B)))
18	17	imbi2d 674	. . . . . . . . . . . 12 |- (x = B -> ((C e. On -> (C +o suc A) C_ (C +o x)) <-> (C e. On -> (C +o suc A) C_ (C +o B))))
19		ssid 2634	. . . . . . . . . . . . 13 |- (C +o suc A) C_ (C +o suc A)
20	19	a1i12 9	. . . . . . . . . . . 12 |- (suc A e. On -> (C e. On -> (C +o suc A) C_ (C +o suc A)))
21		oasuc 5208	. . . . . . . . . . . . . . . . . 18 |- ((C e. On /\ y e. On) -> (C +o suc y) = suc (C +o y))
22	21	ancoms 484	. . . . . . . . . . . . . . . . 17 |- ((y e. On /\ C e. On) -> (C +o suc y) = suc (C +o y))
23	22	sseq2d 2645	. . . . . . . . . . . . . . . 16 |- ((y e. On /\ C e. On) -> ((C +o suc A) C_ (C +o suc y) <-> (C +o suc A) C_ suc (C +o y)))
24		sssucid 3742	. . . . . . . . . . . . . . . . 17 |- (C +o y) C_ suc (C +o y)
25		sstr2 2623	. . . . . . . . . . . . . . . . 17 |- ((C +o suc A) C_ (C +o y) -> ((C +o y) C_ suc (C +o y) -> (C +o suc A) C_ suc (C +o y)))
26	24, 25	mpi 55	. . . . . . . . . . . . . . . 16 |- ((C +o suc A) C_ (C +o y) -> (C +o suc A) C_ suc (C +o y))
27	23, 26	syl5bir 227	. . . . . . . . . . . . . . 15 |- ((y e. On /\ C e. On) -> ((C +o suc A) C_ (C +o y) -> (C +o suc A) C_ (C +o suc y)))
28	27	ex 402	. . . . . . . . . . . . . 14 |- (y e. On -> (C e. On -> ((C +o suc A) C_ (C +o y) -> (C +o suc A) C_ (C +o suc y))))
29	28	ad2antrr 440	. . . . . . . . . . . . 13 |- (((y e. On /\ suc A e. On) /\ suc A C_ y) -> (C e. On -> ((C +o suc A) C_ (C +o y) -> (C +o suc A) C_ (C +o suc y))))
30	29	a2d 16	. . . . . . . . . . . 12 |- (((y e. On /\ suc A e. On) /\ suc A C_ y) -> ((C e. On -> (C +o suc A) C_ (C +o y)) -> (C e. On -> (C +o suc A) C_ (C +o suc y))))
31		sucelon 3898	. . . . . . . . . . . . . . . . . . . 20 |- (A e. On <-> suc A e. On)
32		sucssel 3763	. . . . . . . . . . . . . . . . . . . 20 |- (A e. On -> (suc A C_ x -> A e. x))
33	31, 32	sylbir 218	. . . . . . . . . . . . . . . . . . 19 |- (suc A e. On -> (suc A C_ x -> A e. x))
34		limsuc 3933	. . . . . . . . . . . . . . . . . . . 20 |- (Lim x -> (A e. x <-> suc A e. x))
35	34	biimpd 170	. . . . . . . . . . . . . . . . . . 19 |- (Lim x -> (A e. x -> suc A e. x))
36	33, 35	sylan9r 519	. . . . . . . . . . . . . . . . . 18 |- ((Lim x /\ suc A e. On) -> (suc A C_ x -> suc A e. x))
37	36	imp 377	. . . . . . . . . . . . . . . . 17 |- (((Lim x /\ suc A e. On) /\ suc A C_ x) -> suc A e. x)
38		opreq2 4890	. . . . . . . . . . . . . . . . . 18 |- (y = suc A -> (C +o y) = (C +o suc A))
39	38	ssiun2s 3297	. . . . . . . . . . . . . . . . 17 |- (suc A e. x -> (C +o suc A) C_ U_y e. x (C +o y))
40	37, 39	syl 12	. . . . . . . . . . . . . . . 16 |- (((Lim x /\ suc A e. On) /\ suc A C_ x) -> (C +o suc A) C_ U_y e. x (C +o y))
41	40	adantr 425	. . . . . . . . . . . . . . 15 |- ((((Lim x /\ suc A e. On) /\ suc A C_ x) /\ C e. On) -> (C +o suc A) C_ U_y e. x (C +o y))
42		visset 2295	. . . . . . . . . . . . . . . . . . 19 |- x e. _V
43		oalim 5212	. . . . . . . . . . . . . . . . . . 19 |- ((C e. On /\ (x e. _V /\ Lim x)) -> (C +o x) = U_y e. x (C +o y))
44	42, 43	mpanr1 774	. . . . . . . . . . . . . . . . . 18 |- ((C e. On /\ Lim x) -> (C +o x) = U_y e. x (C +o y))
45	44	ancoms 484	. . . . . . . . . . . . . . . . 17 |- ((Lim x /\ C e. On) -> (C +o x) = U_y e. x (C +o y))
46	45	adantlr 429	. . . . . . . . . . . . . . . 16 |- (((Lim x /\ suc A e. On) /\ C e. On) -> (C +o x) = U_y e. x (C +o y))
47	46	adantlr 429	. . . . . . . . . . . . . . 15 |- ((((Lim x /\ suc A e. On) /\ suc A C_ x) /\ C e. On) -> (C +o x) = U_y e. x (C +o y))
48	41, 47	sseqtr4d 2654	. . . . . . . . . . . . . 14 |- ((((Lim x /\ suc A e. On) /\ suc A C_ x) /\ C e. On) -> (C +o suc A) C_ (C +o x))
49	48	ex 402	. . . . . . . . . . . . 13 |- (((Lim x /\ suc A e. On) /\ suc A C_ x) -> (C e. On -> (C +o suc A) C_ (C +o x)))
50	49	a1d 15	. . . . . . . . . . . 12 |- (((Lim x /\ suc A e. On) /\ suc A C_ x) -> (A.y e. x (suc A C_ y -> (C e. On -> (C +o suc A) C_ (C +o y))) -> (C e. On -> (C +o suc A) C_ (C +o x))))
51	9, 12, 15, 18, 20, 30, 50	tfindsg 3944	. . . . . . . . . . 11 |- (((B e. On /\ suc A e. On) /\ suc A C_ B) -> (C e. On -> (C +o suc A) C_ (C +o B)))
52	51	exp31 407	. . . . . . . . . 10 |- (B e. On -> (suc A e. On -> (suc A C_ B -> (C e. On -> (C +o suc A) C_ (C +o B)))))
53	52, 31	syl5ib 223	. . . . . . . . 9 |- (B e. On -> (A e. On -> (suc A C_ B -> (C e. On -> (C +o suc A) C_ (C +o B)))))
54	53	com4r 45	. . . . . . . 8 |- (C e. On -> (B e. On -> (A e. On -> (suc A C_ B -> (C +o suc A) C_ (C +o B)))))
55	54	imp31 389	. . . . . . 7 |- (((C e. On /\ B e. On) /\ A e. On) -> (suc A C_ B -> (C +o suc A) C_ (C +o B)))
56		oasuc 5208	. . . . . . . . . 10 |- ((C e. On /\ A e. On) -> (C +o suc A) = suc (C +o A))
57	56	sseq1d 2644	. . . . . . . . 9 |- ((C e. On /\ A e. On) -> ((C +o suc A) C_ (C +o B) <-> suc (C +o A) C_ (C +o B)))
58		oprex 4907	. . . . . . . . . 10 |- (C +o A) e. _V
59		sucssel 3763	. . . . . . . . . 10 |- ((C +o A) e. _V -> (suc (C +o A) C_ (C +o B) -> (C +o A) e. (C +o B)))
60	58, 59	ax-mp 7	. . . . . . . . 9 |- (suc (C +o A) C_ (C +o B) -> (C +o A) e. (C +o B))
61	57, 60	syl6bi 231	. . . . . . . 8 |- ((C e. On /\ A e. On) -> ((C +o suc A) C_ (C +o B) -> (C +o A) e. (C +o B)))
62	61	adantlr 429	. . . . . . 7 |- (((C e. On /\ B e. On) /\ A e. On) -> ((C +o suc A) C_ (C +o B) -> (C +o A) e. (C +o B)))
63	6, 55, 62	3syld 31	. . . . . 6 |- (((C e. On /\ B e. On) /\ A e. On) -> (A e. B -> (C +o A) e. (C +o B)))
64	63	imp 377	. . . . 5 |- ((((C e. On /\ B e. On) /\ A e. On) /\ A e. B) -> (C +o A) e. (C +o B))
65	64	an1rs 547	. . . 4 |- ((((C e. On /\ B e. On) /\ A e. B) /\ A e. On) -> (C +o A) e. (C +o B))
66	2, 65	mpdan 768	. . 3 |- (((C e. On /\ B e. On) /\ A e. B) -> (C +o A) e. (C +o B))
67	66	ex 402	. 2 |- ((C e. On /\ B e. On) -> (A e. B -> (C +o A) e. (C +o B)))
68	67	ancoms 484	1 |- ((B e. On /\ C e. On) -> (A e. B -> (C +o A) e. (C +o B)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  U_ciun 3255  Ord word 3656  Oncon0 3657  Lim wlim 3658  suc csuc 3659  (class class class)co 4884   +o coa 5174

This theorem is referenced by:  oaord 5228  oaass 5243  odi 5258  nnaordi 5289

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-oadd 5179

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