Theorem oaordi 8531

Description: Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)

Assertion

Ref	Expression
oaordi	⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))

Proof of Theorem oaordi

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onelon 6386	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
2	1	adantll 726	. . . 4 ⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
3		eloni 6371	. . . . . . . . 9 ⊢ (𝐵 ∈ On → Ord 𝐵)
4		ordsucss 7814	. . . . . . . . 9 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
5	3, 4	syl 18	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
6	5	ad2antlr 739	. . . . . . 7 ⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
7		onsucb 7813	. . . . . . . . . 10 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
8		oveq2 7419	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝐴 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝐴))
9	8	sseq2d 3977	. . . . . . . . . . . . 13 ⊢ (𝑥 = suc 𝐴 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴)))
10	9	imbi2d 343	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝐴 → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴))))
11		oveq2 7419	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o 𝑦))
12	11	sseq2d 3977	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦)))
13	12	imbi2d 343	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦))))
14		oveq2 7419	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝑦))
15	14	sseq2d 3977	. . . . . . . . . . . . 13 ⊢ (𝑥 = suc 𝑦 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))
16	15	imbi2d 343	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝑦 → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
17		oveq2 7419	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → (𝐶 +o 𝑥) = (𝐶 +o 𝐵))
18	17	sseq2d 3977	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
19	18	imbi2d 343	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵))))
20		ssid 3967	. . . . . . . . . . . . 13 ⊢ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴)
21	20	2a1i 12	. . . . . . . . . . . 12 ⊢ (suc 𝐴 ∈ On → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴)))
22		sssucid 6444	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 +o 𝑦) ⊆ suc (𝐶 +o 𝑦)
23		sstr2 3952	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → ((𝐶 +o 𝑦) ⊆ suc (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦)))
24	22, 23	mpi 21	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦))
25		oasuc 8509	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
26	25	ancoms 463	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ On ∧ 𝐶 ∈ On) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
27	26	sseq2d 3977	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦) ↔ (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦)))
28	24, 27	imbitrrid 249	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))
29	28	ex 417	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ On → (𝐶 ∈ On → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
30	29	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑦) → (𝐶 ∈ On → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
31	30	a2d 30	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑦) → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦)) → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
32		sucssel 6459	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ On → (suc 𝐴 ⊆ 𝑥 → 𝐴 ∈ 𝑥))
33	7, 32	sylbir 238	. . . . . . . . . . . . . . . . . . 19 ⊢ (suc 𝐴 ∈ On → (suc 𝐴 ⊆ 𝑥 → 𝐴 ∈ 𝑥))
34		limsuc 7845	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥))
35	34	biimpd 232	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → suc 𝐴 ∈ 𝑥))
36	33, 35	sylan9r 517	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim 𝑥 ∧ suc 𝐴 ∈ On) → (suc 𝐴 ⊆ 𝑥 → suc 𝐴 ∈ 𝑥))
37	36	imp 411	. . . . . . . . . . . . . . . . 17 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) → suc 𝐴 ∈ 𝑥)
38		oveq2 7419	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = suc 𝐴 → (𝐶 +o 𝑦) = (𝐶 +o suc 𝐴))
39	38	ssiun2s 5017	. . . . . . . . . . . . . . . . 17 ⊢ (suc 𝐴 ∈ 𝑥 → (𝐶 +o suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
40	37, 39	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) → (𝐶 +o suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
41	40	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) ∧ 𝐶 ∈ On) → (𝐶 +o suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
42		vex 3467	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑥 ∈ V
43		oalim 8517	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
44	42, 43	mpanr1 715	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
45	44	ancoms 463	. . . . . . . . . . . . . . . . 17 ⊢ ((Lim 𝑥 ∧ 𝐶 ∈ On) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
46	45	adantlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ 𝐶 ∈ On) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
47	46	adantlr 727	. . . . . . . . . . . . . . 15 ⊢ ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) ∧ 𝐶 ∈ On) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
48	41, 47	sseqtrrd 3982	. . . . . . . . . . . . . 14 ⊢ ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) ∧ 𝐶 ∈ On) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥))
49	48	ex 417	. . . . . . . . . . . . 13 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)))
50	49	a1d 26	. . . . . . . . . . . 12 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) → (∀𝑦 ∈ 𝑥 (suc 𝐴 ⊆ 𝑦 → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦))) → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥))))
51	10, 13, 16, 19, 21, 31, 50	tfindsg 7857	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝐵) → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
52	51	exp31 424	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (suc 𝐴 ∈ On → (suc 𝐴 ⊆ 𝐵 → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
53	7, 52	biimtrid 245	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (suc 𝐴 ⊆ 𝐵 → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
54	53	com4r 95	. . . . . . . 8 ⊢ (𝐶 ∈ On → (𝐵 ∈ On → (𝐴 ∈ On → (suc 𝐴 ⊆ 𝐵 → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
55	54	imp31 422	. . . . . . 7 ⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (suc 𝐴 ⊆ 𝐵 → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
56		oasuc 8509	. . . . . . . . . 10 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 +o suc 𝐴) = suc (𝐶 +o 𝐴))
57	56	sseq1d 3976	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) ↔ suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
58		ovex 7444	. . . . . . . . . 10 ⊢ (𝐶 +o 𝐴) ∈ V
59		sucssel 6459	. . . . . . . . . 10 ⊢ ((𝐶 +o 𝐴) ∈ V → (suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
60	58, 59	ax-mp 5	. . . . . . . . 9 ⊢ (suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
61	57, 60	biimtrdi 256	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
62	61	adantlr 727	. . . . . . 7 ⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
63	6, 55, 62	3syld 61	. . . . . 6 ⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
64	63	imp 411	. . . . 5 ⊢ ((((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) ∧ 𝐴 ∈ 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
65	64	an32s 664	. . . 4 ⊢ ((((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) ∧ 𝐴 ∈ On) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
66	2, 65	mpdan 699	. . 3 ⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
67	66	ex 417	. 2 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
68	67	ancoms 463	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   ⊆ wss 3913  ∪ ciun 4960  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363  (class class class)co 7411   +_o coa 8450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-oadd 8457

This theorem is referenced by:  oaord  8532  oaass  8546  odi  8564  onexomgt  43860  onexoegt  43863  oaltublim  43909  oaordi3  43910  oacl2g  43949  tfsconcatfv2  43959  tfsconcatrn  43961  tfsconcatrev  43967  ofoafg  43973  oaun3lem1  43993  oaun3lem2  43994  oadif1  43999  naddwordnexlem0  44015  naddwordnexlem3  44018  naddwordnexlem4  44020  oaltom  44023