Theorem oaordi 8513

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 6360	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
2	1	adantll 714	. . . 4 ⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
3		eloni 6345	. . . . . . . . 9 ⊢ (𝐵 ∈ On → Ord 𝐵)
4		ordsucss 7796	. . . . . . . . 9 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
5	3, 4	syl 17	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
6	5	ad2antlr 727	. . . . . . 7 ⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
7		onsucb 7795	. . . . . . . . . 10 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
8		oveq2 7398	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝐴 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝐴))
9	8	sseq2d 3982	. . . . . . . . . . . . 13 ⊢ (𝑥 = suc 𝐴 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴)))
10	9	imbi2d 340	. . . . . . . . . . . 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 7398	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o 𝑦))
12	11	sseq2d 3982	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦)))
13	12	imbi2d 340	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦))))
14		oveq2 7398	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝑦))
15	14	sseq2d 3982	. . . . . . . . . . . . 13 ⊢ (𝑥 = suc 𝑦 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))
16	15	imbi2d 340	. . . . . . . . . . . 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 7398	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → (𝐶 +o 𝑥) = (𝐶 +o 𝐵))
18	17	sseq2d 3982	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
19	18	imbi2d 340	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵))))
20		ssid 3972	. . . . . . . . . . . . 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 6417	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 +o 𝑦) ⊆ suc (𝐶 +o 𝑦)
23		sstr2 3956	. . . . . . . . . . . . . . . . 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 20	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦))
25		oasuc 8491	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
26	25	ancoms 458	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ On ∧ 𝐶 ∈ On) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
27	26	sseq2d 3982	. . . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))
29	28	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ On → (𝐶 ∈ On → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
30	29	ad2antrr 726	. . . . . . . . . . . . 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 29	. . . . . . . . . . . 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 6432	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ On → (suc 𝐴 ⊆ 𝑥 → 𝐴 ∈ 𝑥))
33	7, 32	sylbir 235	. . . . . . . . . . . . . . . . . . 19 ⊢ (suc 𝐴 ∈ On → (suc 𝐴 ⊆ 𝑥 → 𝐴 ∈ 𝑥))
34		limsuc 7828	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥))
35	34	biimpd 229	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 → suc 𝐴 ∈ 𝑥))
36	33, 35	sylan9r 508	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim 𝑥 ∧ suc 𝐴 ∈ On) → (suc 𝐴 ⊆ 𝑥 → suc 𝐴 ∈ 𝑥))
37	36	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) → suc 𝐴 ∈ 𝑥)
38		oveq2 7398	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = suc 𝐴 → (𝐶 +o 𝑦) = (𝐶 +o suc 𝐴))
39	38	ssiun2s 5015	. . . . . . . . . . . . . . . . 17 ⊢ (suc 𝐴 ∈ 𝑥 → (𝐶 +o suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
40	37, 39	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) → (𝐶 +o suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
41	40	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) ∧ 𝐶 ∈ On) → (𝐶 +o suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
42		vex 3454	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑥 ∈ V
43		oalim 8499	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
44	42, 43	mpanr1 703	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
45	44	ancoms 458	. . . . . . . . . . . . . . . . 17 ⊢ ((Lim 𝑥 ∧ 𝐶 ∈ On) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
46	45	adantlr 715	. . . . . . . . . . . . . . . 16 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ 𝐶 ∈ On) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
47	46	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) ∧ 𝐶 ∈ On) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
48	41, 47	sseqtrrd 3987	. . . . . . . . . . . . . 14 ⊢ ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) ∧ 𝐶 ∈ On) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥))
49	48	ex 412	. . . . . . . . . . . . 13 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)))
50	49	a1d 25	. . . . . . . . . . . 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 7840	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝐵) → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
52	51	exp31 419	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (suc 𝐴 ∈ On → (suc 𝐴 ⊆ 𝐵 → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
53	7, 52	biimtrid 242	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (suc 𝐴 ⊆ 𝐵 → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
54	53	com4r 94	. . . . . . . 8 ⊢ (𝐶 ∈ On → (𝐵 ∈ On → (𝐴 ∈ On → (suc 𝐴 ⊆ 𝐵 → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
55	54	imp31 417	. . . . . . 7 ⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (suc 𝐴 ⊆ 𝐵 → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
56		oasuc 8491	. . . . . . . . . 10 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 +o suc 𝐴) = suc (𝐶 +o 𝐴))
57	56	sseq1d 3981	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) ↔ suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
58		ovex 7423	. . . . . . . . . 10 ⊢ (𝐶 +o 𝐴) ∈ V
59		sucssel 6432	. . . . . . . . . 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 253	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
62	61	adantlr 715	. . . . . . 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 60	. . . . . 6 ⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
64	63	imp 406	. . . . 5 ⊢ ((((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) ∧ 𝐴 ∈ 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
65	64	an32s 652	. . . 4 ⊢ ((((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) ∧ 𝐴 ∈ On) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
66	2, 65	mpdan 687	. . 3 ⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
67	66	ex 412	. 2 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
68	67	ancoms 458	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   ⊆ wss 3917  ∪ ciun 4958  Ord word 6334  Oncon0 6335  Lim wlim 6336  suc csuc 6337  (class class class)co 7390   +_o coa 8434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-oadd 8441

This theorem is referenced by:  oaord  8514  oaass  8528  odi  8546  onexomgt  43237  onexoegt  43240  oaltublim  43286  oaordi3  43287  oacl2g  43326  tfsconcatfv2  43336  tfsconcatrn  43338  tfsconcatrev  43344  ofoafg  43350  oaun3lem1  43370  oaun3lem2  43371  oadif1  43376  naddwordnexlem0  43392  naddwordnexlem3  43395  naddwordnexlem4  43397  oaltom  43401