Theorem oaordi 8470

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 6337	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
2	1	adantll 715	. . . 4 ⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
3		eloni 6322	. . . . . . . . 9 ⊢ (𝐵 ∈ On → Ord 𝐵)
4		ordsucss 7758	. . . . . . . . 9 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
5	3, 4	syl 17	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
6	5	ad2antlr 728	. . . . . . 7 ⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
7		onsucb 7757	. . . . . . . . . 10 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
8		oveq2 7364	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝐴 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝐴))
9	8	sseq2d 3949	. . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o 𝑦))
12	11	sseq2d 3949	. . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝑦))
15	14	sseq2d 3949	. . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → (𝐶 +o 𝑥) = (𝐶 +o 𝐵))
18	17	sseq2d 3949	. . . . . . . . . . . . 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 3939	. . . . . . . . . . . . 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 6394	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 +o 𝑦) ⊆ suc (𝐶 +o 𝑦)
23		sstr2 3924	. . . . . . . . . . . . . . . . 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 8448	. . . . . . . . . . . . . . . . . 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 3949	. . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . 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 6409	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ On → (suc 𝐴 ⊆ 𝑥 → 𝐴 ∈ 𝑥))
33	7, 32	sylbir 235	. . . . . . . . . . . . . . . . . . 19 ⊢ (suc 𝐴 ∈ On → (suc 𝐴 ⊆ 𝑥 → 𝐴 ∈ 𝑥))
34		limsuc 7789	. . . . . . . . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = suc 𝐴 → (𝐶 +o 𝑦) = (𝐶 +o suc 𝐴))
39	38	ssiun2s 4980	. . . . . . . . . . . . . . . . 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 3431	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑥 ∈ V
43		oalim 8456	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
44	42, 43	mpanr1 704	. . . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . 16 ⊢ (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ 𝐶 ∈ On) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
47	46	adantlr 716	. . . . . . . . . . . . . . 15 ⊢ ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) ∧ 𝐶 ∈ On) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦))
48	41, 47	sseqtrrd 3954	. . . . . . . . . . . . . 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 7801	. . . . . . . . . . 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 8448	. . . . . . . . . 10 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 +o suc 𝐴) = suc (𝐶 +o 𝐴))
57	56	sseq1d 3948	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) ↔ suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
58		ovex 7389	. . . . . . . . . 10 ⊢ (𝐶 +o 𝐴) ∈ V
59		sucssel 6409	. . . . . . . . . 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 716	. . . . . . 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 653	. . . 4 ⊢ ((((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) ∧ 𝐴 ∈ On) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
66	2, 65	mpdan 688	. . 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 1542   ∈ wcel 2114  ∀wral 3049  Vcvv 3427   ⊆ wss 3885  ∪ ciun 4923  Ord word 6311  Oncon0 6312  Lim wlim 6313  suc csuc 6314  (class class class)co 7356   +_o coa 8391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-oprab 7360  df-mpo 7361  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-oadd 8398

This theorem is referenced by:  oaord  8471  oaass  8485  odi  8503  onexomgt  43657  onexoegt  43660  oaltublim  43706  oaordi3  43707  oacl2g  43746  tfsconcatfv2  43756  tfsconcatrn  43758  tfsconcatrev  43764  ofoafg  43770  oaun3lem1  43790  oaun3lem2  43791  oadif1  43796  naddwordnexlem0  43812  naddwordnexlem3  43815  naddwordnexlem4  43817  oaltom  43820