Theorem oewordri 7826

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 6801	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 ∅))
2		oveq2 6801	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ↑𝑜 𝑥) = (𝐵 ↑𝑜 ∅))
3	1, 2	sseq12d 3783	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥) ↔ (𝐴 ↑𝑜 ∅) ⊆ (𝐵 ↑𝑜 ∅)))
4		oveq2 6801	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝑦))
5		oveq2 6801	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ↑𝑜 𝑥) = (𝐵 ↑𝑜 𝑦))
6	4, 5	sseq12d 3783	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥) ↔ (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦)))
7		oveq2 6801	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 suc 𝑦))
8		oveq2 6801	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ↑𝑜 𝑥) = (𝐵 ↑𝑜 suc 𝑦))
9	7, 8	sseq12d 3783	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥) ↔ (𝐴 ↑𝑜 suc 𝑦) ⊆ (𝐵 ↑𝑜 suc 𝑦)))
10		oveq2 6801	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝐶))
11		oveq2 6801	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 ↑𝑜 𝑥) = (𝐵 ↑𝑜 𝐶))
12	10, 11	sseq12d 3783	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥) ↔ (𝐴 ↑𝑜 𝐶) ⊆ (𝐵 ↑𝑜 𝐶)))
13		onelon 5891	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
14		oe0 7756	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜)
15	13, 14	syl 17	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑𝑜 ∅) = 1𝑜)
16		oe0 7756	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 ↑𝑜 ∅) = 1𝑜)
17	16	adantr 466	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐵 ↑𝑜 ∅) = 1𝑜)
18	15, 17	eqtr4d 2808	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑𝑜 ∅) = (𝐵 ↑𝑜 ∅))
19		eqimss 3806	. . . . 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 468	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐵 ∈ On)
22		onelss 5909	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
23	22	imp 393	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ 𝐵)
24	13, 21, 23	jca31 504	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵))
25		oecl 7771	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 𝑦) ∈ On)
26	25	3adant2 1125	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 𝑦) ∈ On)
27		oecl 7771	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑𝑜 𝑦) ∈ On)
28	27	3adant1 1124	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑𝑜 𝑦) ∈ On)
29		simp1 1130	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
30		omwordri 7806	. . . . . . . . . . . . 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 1476	. . . . . . . . . . . 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 393	. . . . . . . . . . 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 695	. . . . . . . . . 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 7805	. . . . . . . . . . . . 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 1515	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ⊆ 𝐵 → ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐵)))
36	35	imp 393	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐴) ⊆ ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐵))
37	36	adantrr 696	. . . . . . . . . 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 3762	. . . . . . . . 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 7761	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 suc 𝑦) = ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴))
40	39	3adant2 1125	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 suc 𝑦) = ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴))
41	40	adantr 466	. . . . . . . . 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 7761	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑𝑜 suc 𝑦) = ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐵))
43	42	3adant1 1124	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑𝑜 suc 𝑦) = ((𝐵 ↑𝑜 𝑦) ·𝑜 𝐵))
44	43	adantr 466	. . . . . . . . 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 3797	. . . . . . . 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 1450	. . . . . . 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 87	. . . . . 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 410	. . . . 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 34	. . . 4 ⊢ (𝑦 ∈ On → ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → (𝐴 ↑𝑜 suc 𝑦) ⊆ (𝐵 ↑𝑜 suc 𝑦))))
50		vex 3354	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
51		limelon 5931	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
52	50, 51	mpan 670	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
53		0ellim 5930	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
54		oe0m1 7755	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑𝑜 𝑥) = ∅))
55	54	biimpa 462	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ ∅ ∈ 𝑥) → (∅ ↑𝑜 𝑥) = ∅)
56	52, 53, 55	syl2anc 573	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (∅ ↑𝑜 𝑥) = ∅)
57		0ss 4116	. . . . . . . . . 10 ⊢ ∅ ⊆ (𝐵 ↑𝑜 𝑥)
58	56, 57	syl6eqss 3804	. . . . . . . . 9 ⊢ (Lim 𝑥 → (∅ ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥))
59		oveq1 6800	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝑥) = (∅ ↑𝑜 𝑥))
60	59	sseq1d 3781	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ((𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥) ↔ (∅ ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥)))
61	58, 60	syl5ibr 236	. . . . . . . 8 ⊢ (𝐴 = ∅ → (Lim 𝑥 → (𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥)))
62	61	adantl 467	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ 𝐴 = ∅) → (Lim 𝑥 → (𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥)))
63	62	a1dd 50	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ 𝐴 = ∅) → (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → (𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥))))
64		ss2iun 4670	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 ↑𝑜 𝑦))
65		oelim 7768	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
66	50, 65	mpanlr1 686	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
67	66	an32s 631	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
68	67	adantllr 698	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
69	21	anim1i 602	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ Lim 𝑥) → (𝐵 ∈ On ∧ Lim 𝑥))
70		ne0i 4069	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ ∅)
71		on0eln0 5923	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
72	70, 71	syl5ibr 236	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → ∅ ∈ 𝐵))
73	72	imp 393	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → ∅ ∈ 𝐵)
74	73	adantr 466	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ Lim 𝑥) → ∅ ∈ 𝐵)
75		oelim 7768	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → (𝐵 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑𝑜 𝑦))
76	50, 75	mpanlr1 686	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝐵 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑𝑜 𝑦))
77	69, 74, 76	syl2anc 573	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ Lim 𝑥) → (𝐵 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑𝑜 𝑦))
78	77	adantlr 694	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑𝑜 𝑦))
79	78	adantlll 697	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑𝑜 𝑦))
80	68, 79	sseq12d 3783	. . . . . . . 8 ⊢ ((((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)))
81	64, 80	syl5ibr 236	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → (𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥)))
82	81	ex 397	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) → (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → (𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥))))
83	63, 82	oe0lem 7747	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) → (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → (𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥))))
84	13	ancri 539	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)))
85	83, 84	syl11 33	. . . 4 ⊢ (Lim 𝑥 → ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (∀𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦) ⊆ (𝐵 ↑𝑜 𝑦) → (𝐴 ↑𝑜 𝑥) ⊆ (𝐵 ↑𝑜 𝑥))))
86	3, 6, 9, 12, 20, 49, 85	tfinds3 7211	. . 3 ⊢ (𝐶 ∈ On → ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑𝑜 𝐶) ⊆ (𝐵 ↑𝑜 𝐶)))
87	86	expd 400	. 2 ⊢ (𝐶 ∈ On → (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (𝐴 ↑𝑜 𝐶) ⊆ (𝐵 ↑𝑜 𝐶))))
88	87	impcom 394	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑𝑜 𝐶) ⊆ (𝐵 ↑𝑜 𝐶)))

Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ≠ wne 2943  ∀wral 3061  Vcvv 3351   ⊆ wss 3723  ∅c0 4063  ∪ ciun 4654  Oncon0 5866  Lim wlim 5867  suc csuc 5868  (class class class)co 6793  1_𝑜c1o 7706   ·_𝑜 comu 7711   ↑_𝑜 coe 7712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-omul 7718  df-oexp 7719

This theorem is referenced by:  oeordsuc  7828