Theorem oewordri 7559

Description: Weak ordering property of ordinal exponentiation. Proposition 8.35 of [TakeutiZaring] p. 68. (Contributed by NM, 6-Jan-2005.)

Assertion

Ref	Expression
oewordri	⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑𝑜 𝐶) ⊆ (𝐵 ↑𝑜 𝐶)))

Proof of Theorem oewordri

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

Step	Hyp	Ref	Expression
1		oveq2 6557	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 ∅))
2		oveq2 6557	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ↑𝑜 𝑥) = (𝐵 ↑𝑜 ∅))
3	1, 2	sseq12d 3597	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥) ↔ (𝐴 ↑𝑜 ∅) ⊆ (𝐵 ↑𝑜 ∅)))
4		oveq2 6557	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝑦))
5		oveq2 6557	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ↑𝑜 𝑥) = (𝐵 ↑𝑜 𝑦))
6	4, 5	sseq12d 3597	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥) ↔ (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦)))
7		oveq2 6557	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 suc 𝑦))
8		oveq2 6557	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ↑𝑜 𝑥) = (𝐵 ↑𝑜 suc 𝑦))
9	7, 8	sseq12d 3597	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥) ↔ (𝐴 ↑𝑜 suc 𝑦) ⊆ (𝐵 ↑𝑜 suc 𝑦)))
10		oveq2 6557	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝐶))
11		oveq2 6557	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 ↑𝑜 𝑥) = (𝐵 ↑𝑜 𝐶))
12	10, 11	sseq12d 3597	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥) ↔ (𝐴 ↑𝑜 𝐶) ⊆ (𝐵 ↑𝑜 𝐶)))
13		onelon 5665	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
14		oe0 7489	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜)
15	13, 14	syl 17	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑𝑜 ∅) = 1𝑜)
16		oe0 7489	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 ↑𝑜 ∅) = 1𝑜)
17	16	adantr 480	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐵 ↑𝑜 ∅) = 1𝑜)
18	15, 17	eqtr4d 2647	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑𝑜 ∅) = (𝐵 ↑𝑜 ∅))
19		eqimss 3620	. . . . 5 ⊢ ((𝐴 ↑𝑜 ∅) = (𝐵 ↑𝑜 ∅) → (𝐴 ↑𝑜 ∅) ⊆ (𝐵 ↑𝑜 ∅))
20	18, 19	syl 17	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑𝑜 ∅) ⊆ (𝐵 ↑𝑜 ∅))
21		simpl 472	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐵 ∈ On)
22		onelss 5683	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
23	22	imp 444	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ 𝐵)
24	13, 21, 23	jca31 555	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵))
25		oecl 7504	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 𝑦) ∈ On)
26	25	3adant2 1073	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 𝑦) ∈ On)
27		oecl 7504	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑𝑜 𝑦) ∈ On)
28	27	3adant1 1072	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑𝑜 𝑦) ∈ On)
29		simp1 1054	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
30		omwordri 7539	. . . . . . . . . . . . 13 ⊢ (((𝐴 ↑𝑜 𝑦) ∈ On ∧ (𝐵 ↑𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐴)))
31	26, 28, 29, 30	syl3anc 1318	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐴)))
32	31	imp 444	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦)) → ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐴))
33	32	adantrl 748	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦))) → ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐴))
34		omwordi 7538	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐵 ↑𝑜 𝑦) ∈ On) → (𝐴 ⊆ 𝐵 → ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐵)))
35	28, 34	syld3an3 1363	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ⊆ 𝐵 → ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐵)))
36	35	imp 444	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐵))
37	36	adantrr 749	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦))) → ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐵))
38	33, 37	sstrd 3578	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦))) → ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐵))
39		oesuc 7494	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 suc 𝑦) = ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴))
40	39	3adant2 1073	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 suc 𝑦) = ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴))
41	40	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦))) → (𝐴 ↑𝑜 suc 𝑦) = ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴))
42		oesuc 7494	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑𝑜 suc 𝑦) = ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐵))
43	42	3adant1 1072	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑𝑜 suc 𝑦) = ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐵))
44	43	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦))) → (𝐵 ↑𝑜 suc 𝑦) = ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐵))
45	38, 41, 44	3sstr4d 3611	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦))) → (𝐴 ↑𝑜 suc 𝑦) ⊆ (𝐵 ↑𝑜 suc 𝑦))
46	45	exp520 1280	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (𝐴 ⊆ 𝐵 → ((𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → (𝐴 ↑𝑜 suc 𝑦) ⊆ (𝐵 ↑𝑜 suc 𝑦))))))
47	46	com3r 85	. . . . . 6 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → (𝐴 ⊆ 𝐵 → ((𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → (𝐴 ↑𝑜 suc 𝑦) ⊆ (𝐵 ↑𝑜 suc 𝑦))))))
48	47	imp4c 615	. . . . 5 ⊢ (𝑦 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → (𝐴 ↑𝑜 suc 𝑦) ⊆ (𝐵 ↑𝑜 suc 𝑦))))
49	24, 48	syl5 33	. . . 4 ⊢ (𝑦 ∈ On → ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → (𝐴 ↑𝑜 suc 𝑦) ⊆ (𝐵 ↑𝑜 suc 𝑦))))
50	13	ancri 573	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)))
51		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
52		limelon 5705	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
53	51, 52	mpan 702	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
54		0ellim 5704	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
55		oe0m1 7488	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑𝑜 𝑥) = ∅))
56	55	biimpa 500	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ ∅ ∈ 𝑥) → (∅ ↑𝑜 𝑥) = ∅)
57	53, 54, 56	syl2anc 691	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → (∅ ↑𝑜 𝑥) = ∅)
58		0ss 3924	. . . . . . . . . . 11 ⊢ ∅ ⊆ (𝐵 ↑𝑜 𝑥)
59	57, 58	syl6eqss 3618	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (∅ ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥))
60		oveq1 6556	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝑥) = (∅ ↑𝑜 𝑥))
61	60	sseq1d 3595	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ((𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥) ↔ (∅ ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥)))
62	59, 61	syl5ibr 235	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (Lim 𝑥 → (𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥)))
63	62	adantl 481	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ 𝐴 = ∅) → (Lim 𝑥 → (𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥)))
64	63	a1dd 48	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ 𝐴 = ∅) → (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → (𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥))))
65		ss2iun 4472	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 ↑𝑜 𝑦))
66		oelim 7501	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
67	51, 66	mpanlr1 718	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
68	67	an32s 842	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
69	68	adantllr 751	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
70	21	anim1i 590	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ Lim 𝑥) → (𝐵 ∈ On ∧ Lim 𝑥))
71		ne0i 3880	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ ∅)
72		on0eln0 5697	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
73	71, 72	syl5ibr 235	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → ∅ ∈ 𝐵))
74	73	imp 444	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → ∅ ∈ 𝐵)
75	74	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ Lim 𝑥) → ∅ ∈ 𝐵)
76		oelim 7501	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → (𝐵 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑𝑜 𝑦))
77	51, 76	mpanlr1 718	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝐵 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑𝑜 𝑦))
78	70, 75, 77	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ Lim 𝑥) → (𝐵 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑𝑜 𝑦))
79	78	adantlr 747	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑𝑜 𝑦))
80	79	adantlll 750	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑𝑜 𝑦))
81	69, 80	sseq12d 3597	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → ((𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥) ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 ↑𝑜 𝑦)))
82	65, 81	syl5ibr 235	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → (𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥)))
83	82	ex 449	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) → (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → (𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥))))
84	64, 83	oe0lem 7480	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) → (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → (𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥))))
85	84	com12 32	. . . . 5 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) → (∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → (𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥))))
86	50, 85	syl5 33	. . . 4 ⊢ (Lim 𝑥 → ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → (𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥))))
87	3, 6, 9, 12, 20, 49, 86	tfinds3 6956	. . 3 ⊢ (𝐶 ∈ On → ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑𝑜 𝐶) ⊆ (𝐵 ↑𝑜 𝐶)))
88	87	expd 451	. 2 ⊢ (𝐶 ∈ On → (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (𝐴 ↑𝑜 𝐶) ⊆ (𝐵 ↑𝑜 𝐶))))
89	88	impcom 445	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑𝑜 𝐶) ⊆ (𝐵 ↑𝑜 𝐶)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549  1_𝑜c1o 7440   ·_𝑜 comu 7445   ↑_𝑜 coe 7446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-oexp 7453

This theorem is referenced by:  oeordsuc  7561