Theorem oaordi 8464

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 6332	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
2	1	adantll 714	. . . 4 ⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
3		eloni 6317	. . . . . . . . 9 ⊢ (𝐵 ∈ On → Ord 𝐵)
4		ordsucss 7751	. . . . . . . . 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 7750	. . . . . . . . . 10 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
8		oveq2 7357	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝐴 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝐴))
9	8	sseq2d 3968	. . . . . . . . . . . . 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 7357	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o 𝑦))
12	11	sseq2d 3968	. . . . . . . . . . . . 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 7357	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝑦))
15	14	sseq2d 3968	. . . . . . . . . . . . 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 7357	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → (𝐶 +o 𝑥) = (𝐶 +o 𝐵))
18	17	sseq2d 3968	. . . . . . . . . . . . 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 3958	. . . . . . . . . . . . 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 6389	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 +o 𝑦) ⊆ suc (𝐶 +o 𝑦)
23		sstr2 3942	. . . . . . . . . . . . . . . . 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 8442	. . . . . . . . . . . . . . . . . 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 3968	. . . . . . . . . . . . . . . 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 6404	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ On → (suc 𝐴 ⊆ 𝑥 → 𝐴 ∈ 𝑥))
33	7, 32	sylbir 235	. . . . . . . . . . . . . . . . . . 19 ⊢ (suc 𝐴 ∈ On → (suc 𝐴 ⊆ 𝑥 → 𝐴 ∈ 𝑥))
34		limsuc 7782	. . . . . . . . . . . . . . . . . . . 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 7357	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = suc 𝐴 → (𝐶 +o 𝑦) = (𝐶 +o suc 𝐴))
39	38	ssiun2s 4997	. . . . . . . . . . . . . . . . 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 3440	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑥 ∈ V
43		oalim 8450	. . . . . . . . . . . . . . . . . . 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 3973	. . . . . . . . . . . . . 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 7794	. . . . . . . . . . 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 8442	. . . . . . . . . 10 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 +o suc 𝐴) = suc (𝐶 +o 𝐴))
57	56	sseq1d 3967	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) ↔ suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
58		ovex 7382	. . . . . . . . . 10 ⊢ (𝐶 +o 𝐴) ∈ V
59		sucssel 6404	. . . . . . . . . 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 3044  Vcvv 3436   ⊆ wss 3903  ∪ ciun 4941  Ord word 6306  Oncon0 6307  Lim wlim 6308  suc csuc 6309  (class class class)co 7349   +_o coa 8385

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-oadd 8392

This theorem is referenced by:  oaord  8465  oaass  8479  odi  8497  onexomgt  43218  onexoegt  43221  oaltublim  43267  oaordi3  43268  oacl2g  43307  tfsconcatfv2  43317  tfsconcatrn  43319  tfsconcatrev  43325  ofoafg  43331  oaun3lem1  43351  oaun3lem2  43352  oadif1  43357  naddwordnexlem0  43373  naddwordnexlem3  43376  naddwordnexlem4  43378  oaltom  43382