Theorem omordi 8484

Description: Ordering property of ordinal multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63. Lemma 3.15 of [Schloeder] p. 9. (Contributed by NM, 14-Dec-2004.)

Assertion

Ref	Expression
omordi	⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

Proof of Theorem omordi

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

Step	Hyp	Ref	Expression
1		onelon 6332	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
2	1	ex 412	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ∈ On))
3		eleq2 2817	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
4		oveq2 7357	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝐶 ·o 𝑥) = (𝐶 ·o ∅))
5	4	eleq2d 2814	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅)))
6	3, 5	imbi12d 344	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅))))
7		eleq2 2817	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
8		oveq2 7357	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝑦))
9	8	eleq2d 2814	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))
10	7, 9	imbi12d 344	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦))))
11		eleq2 2817	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
12		oveq2 7357	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o suc 𝑦))
13	12	eleq2d 2814	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))
14	11, 13	imbi12d 344	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))
15		eleq2 2817	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
16		oveq2 7357	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝐵))
17	16	eleq2d 2814	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
18	15, 17	imbi12d 344	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
19		noel 4289	. . . . . . . . . . 11 ⊢ ¬ 𝐴 ∈ ∅
20	19	pm2.21i 119	. . . . . . . . . 10 ⊢ (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅))
21	20	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅)))
22		elsuci 6376	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ suc 𝑦 → (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
23		omcl 8454	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ·o 𝑦) ∈ On)
24		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → 𝐶 ∈ On)
25	23, 24	jca 511	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → ((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On))
26		oaword1 8470	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → (𝐶 ·o 𝑦) ⊆ ((𝐶 ·o 𝑦) +o 𝐶))
27	26	sseld 3934	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
28	27	imim2d 57	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶))))
29	28	imp 406	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦))) → (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
30	29	adantrl 716	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
31		oaord1 8469	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
32	31	biimpa 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶))
33		oveq2 7357	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 = 𝑦 → (𝐶 ·o 𝐴) = (𝐶 ·o 𝑦))
34	33	eleq1d 2813	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 = 𝑦 → ((𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶) ↔ (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
35	32, 34	syl5ibrcom 247	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
36	35	adantrr 717	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 = 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
37	30, 36	jaod 859	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
38	25, 37	sylan 580	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
39	22, 38	syl5 34	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
40		omsuc 8444	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ·o suc 𝑦) = ((𝐶 ·o 𝑦) +o 𝐶))
41	40	eleq2d 2814	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦) ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
42	41	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦) ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
43	39, 42	sylibrd 259	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))
44	43	exp43 436	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → (𝑦 ∈ On → (∅ ∈ 𝐶 → ((𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))))
45	44	com12 32	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → ((𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))))
46	45	adantld 490	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → ((𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))))
47	46	impd 410	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))))
48		id 22	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 ∈ On ∧ Lim 𝑥))
49	48	ad2ant2r 747	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) → (𝐶 ∈ On ∧ Lim 𝑥))
50		limsuc 7782	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥))
51	50	biimpa 476	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → suc 𝐴 ∈ 𝑥)
52		oveq2 7357	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = suc 𝐴 → (𝐶 ·o 𝑦) = (𝐶 ·o suc 𝐴))
53	52	ssiun2s 4997	. . . . . . . . . . . . . . . . . 18 ⊢ (suc 𝐴 ∈ 𝑥 → (𝐶 ·o suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 ·o 𝑦))
54	51, 53	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → (𝐶 ·o suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 ·o 𝑦))
55	54	adantll 714	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·o suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 ·o 𝑦))
56		vex 3440	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
57		omlim 8451	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐶 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ·o 𝑦))
58	56, 57	mpanr1 703	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ·o 𝑦))
59	58	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ·o 𝑦))
60	55, 59	sseqtrrd 3973	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·o suc 𝐴) ⊆ (𝐶 ·o 𝑥))
61	49, 60	sylan 580	. . . . . . . . . . . . . 14 ⊢ ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·o suc 𝐴) ⊆ (𝐶 ·o 𝑥))
62		omcl 8454	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·o 𝐴) ∈ On)
63		oaord1 8469	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 ·o 𝐴) ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶)))
64	62, 63	sylan 580	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶)))
65	64	anabss1 666	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶)))
66	65	biimpa 476	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶))
67		omsuc 8444	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·o suc 𝐴) = ((𝐶 ·o 𝐴) +o 𝐶))
68	67	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o suc 𝐴) = ((𝐶 ·o 𝐴) +o 𝐶))
69	66, 68	eleqtrrd 2831	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝐴))
70	69	adantrl 716	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝐴))
71	70	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝐴))
72	61, 71	sseldd 3936	. . . . . . . . . . . . 13 ⊢ ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))
73	72	exp53 447	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → (𝐴 ∈ On → (Lim 𝑥 → (∅ ∈ 𝐶 → (𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))))))
74	73	com13 88	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))))))
75	74	imp4c 423	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))))
76	75	a1dd 50	. . . . . . . . 9 ⊢ (Lim 𝑥 → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)))))
77	6, 10, 14, 18, 21, 47, 76	tfinds3 7798	. . . . . . . 8 ⊢ (𝐵 ∈ On → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
78	77	com23 86	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
79	78	exp4a 431	. . . . . 6 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
80	79	exp4a 431	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (𝐴 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))))
81	2, 80	mpdd 43	. . . 4 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
82	81	com34 91	. . 3 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (∅ ∈ 𝐶 → (𝐶 ∈ On → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
83	82	com24 95	. 2 ⊢ (𝐵 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
84	83	imp31 417	1 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3436   ⊆ wss 3903  ∅c0 4284  ∪ ciun 4941  Oncon0 6307  Lim wlim 6308  suc csuc 6309  (class class class)co 7349   +_o coa 8385   ·_o comu 8386

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-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-oadd 8392  df-omul 8393

This theorem is referenced by:  omord2  8485  omcan  8487  odi  8497  omass  8498  oen0  8504  oeordi  8505  oeordsuc  8512  onexoegt  43221  omord2i  43278  oaomoencom  43294  cantnftermord  43297  omcl2  43310  omltoe  43384