Theorem oeordi 8554

Description: Ordering law for ordinal exponentiation. Proposition 8.33 of [TakeutiZaring] p. 67. (Contributed by NM, 5-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Assertion

Ref	Expression
oeordi	⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))

Proof of Theorem oeordi

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

Step	Hyp	Ref	Expression
1		oveq2 7398	. . . . 5 ⊢ (𝑥 = suc 𝐴 → (𝐶 ↑o 𝑥) = (𝐶 ↑o suc 𝐴))
2	1	eleq2d 2815	. . . 4 ⊢ (𝑥 = suc 𝐴 → ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑥) ↔ (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝐴)))
3	2	imbi2d 340	. . 3 ⊢ (𝑥 = suc 𝐴 → ((𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑥)) ↔ (𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝐴))))
4		oveq2 7398	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐶 ↑o 𝑥) = (𝐶 ↑o 𝑦))
5	4	eleq2d 2815	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑥) ↔ (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦)))
6	5	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑥)) ↔ (𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦))))
7		oveq2 7398	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐶 ↑o 𝑥) = (𝐶 ↑o suc 𝑦))
8	7	eleq2d 2815	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑥) ↔ (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝑦)))
9	8	imbi2d 340	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑥)) ↔ (𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝑦))))
10		oveq2 7398	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐶 ↑o 𝑥) = (𝐶 ↑o 𝐵))
11	10	eleq2d 2815	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑥) ↔ (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))
12	11	imbi2d 340	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑥)) ↔ (𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵))))
13		eldifi 4097	. . . . . . . 8 ⊢ (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
14		oecl 8504	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑o 𝐴) ∈ On)
15	13, 14	sylan 580	. . . . . . 7 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → (𝐶 ↑o 𝐴) ∈ On)
16		om1 8509	. . . . . . 7 ⊢ ((𝐶 ↑o 𝐴) ∈ On → ((𝐶 ↑o 𝐴) ·o 1o) = (𝐶 ↑o 𝐴))
17	15, 16	syl 17	. . . . . 6 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → ((𝐶 ↑o 𝐴) ·o 1o) = (𝐶 ↑o 𝐴))
18		ondif2 8469	. . . . . . . . 9 ⊢ (𝐶 ∈ (On ∖ 2o) ↔ (𝐶 ∈ On ∧ 1o ∈ 𝐶))
19	18	simprbi 496	. . . . . . . 8 ⊢ (𝐶 ∈ (On ∖ 2o) → 1o ∈ 𝐶)
20	19	adantr 480	. . . . . . 7 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → 1o ∈ 𝐶)
21	13	adantr 480	. . . . . . . 8 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → 𝐶 ∈ On)
22		simpr 484	. . . . . . . . 9 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → 𝐴 ∈ On)
23		dif20el 8472	. . . . . . . . . 10 ⊢ (𝐶 ∈ (On ∖ 2o) → ∅ ∈ 𝐶)
24	23	adantr 480	. . . . . . . . 9 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → ∅ ∈ 𝐶)
25		oen0 8553	. . . . . . . . 9 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶 ↑o 𝐴))
26	21, 22, 24, 25	syl21anc 837	. . . . . . . 8 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → ∅ ∈ (𝐶 ↑o 𝐴))
27		omordi 8533	. . . . . . . 8 ⊢ (((𝐶 ∈ On ∧ (𝐶 ↑o 𝐴) ∈ On) ∧ ∅ ∈ (𝐶 ↑o 𝐴)) → (1o ∈ 𝐶 → ((𝐶 ↑o 𝐴) ·o 1o) ∈ ((𝐶 ↑o 𝐴) ·o 𝐶)))
28	21, 15, 26, 27	syl21anc 837	. . . . . . 7 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → (1o ∈ 𝐶 → ((𝐶 ↑o 𝐴) ·o 1o) ∈ ((𝐶 ↑o 𝐴) ·o 𝐶)))
29	20, 28	mpd 15	. . . . . 6 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → ((𝐶 ↑o 𝐴) ·o 1o) ∈ ((𝐶 ↑o 𝐴) ·o 𝐶))
30	17, 29	eqeltrrd 2830	. . . . 5 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → (𝐶 ↑o 𝐴) ∈ ((𝐶 ↑o 𝐴) ·o 𝐶))
31		oesuc 8494	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑o suc 𝐴) = ((𝐶 ↑o 𝐴) ·o 𝐶))
32	13, 31	sylan 580	. . . . 5 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → (𝐶 ↑o suc 𝐴) = ((𝐶 ↑o 𝐴) ·o 𝐶))
33	30, 32	eleqtrrd 2832	. . . 4 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝐴))
34	33	expcom 413	. . 3 ⊢ (𝐴 ∈ On → (𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝐴)))
35		oecl 8504	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ↑o 𝑦) ∈ On)
36	13, 35	sylan 580	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶 ↑o 𝑦) ∈ On)
37		om1 8509	. . . . . . . . . 10 ⊢ ((𝐶 ↑o 𝑦) ∈ On → ((𝐶 ↑o 𝑦) ·o 1o) = (𝐶 ↑o 𝑦))
38	36, 37	syl 17	. . . . . . . . 9 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → ((𝐶 ↑o 𝑦) ·o 1o) = (𝐶 ↑o 𝑦))
39	19	adantr 480	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → 1o ∈ 𝐶)
40	13	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → 𝐶 ∈ On)
41		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → 𝑦 ∈ On)
42	23	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → ∅ ∈ 𝐶)
43		oen0 8553	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶 ↑o 𝑦))
44	40, 41, 42, 43	syl21anc 837	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → ∅ ∈ (𝐶 ↑o 𝑦))
45		omordi 8533	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ On ∧ (𝐶 ↑o 𝑦) ∈ On) ∧ ∅ ∈ (𝐶 ↑o 𝑦)) → (1o ∈ 𝐶 → ((𝐶 ↑o 𝑦) ·o 1o) ∈ ((𝐶 ↑o 𝑦) ·o 𝐶)))
46	40, 36, 44, 45	syl21anc 837	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (1o ∈ 𝐶 → ((𝐶 ↑o 𝑦) ·o 1o) ∈ ((𝐶 ↑o 𝑦) ·o 𝐶)))
47	39, 46	mpd 15	. . . . . . . . 9 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → ((𝐶 ↑o 𝑦) ·o 1o) ∈ ((𝐶 ↑o 𝑦) ·o 𝐶))
48	38, 47	eqeltrrd 2830	. . . . . . . 8 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶 ↑o 𝑦) ∈ ((𝐶 ↑o 𝑦) ·o 𝐶))
49		oesuc 8494	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ↑o suc 𝑦) = ((𝐶 ↑o 𝑦) ·o 𝐶))
50	13, 49	sylan 580	. . . . . . . 8 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶 ↑o suc 𝑦) = ((𝐶 ↑o 𝑦) ·o 𝐶))
51	48, 50	eleqtrrd 2832	. . . . . . 7 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶 ↑o 𝑦) ∈ (𝐶 ↑o suc 𝑦))
52		onsuc 7790	. . . . . . . . 9 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
53		oecl 8504	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ suc 𝑦 ∈ On) → (𝐶 ↑o suc 𝑦) ∈ On)
54	13, 52, 53	syl2an 596	. . . . . . . 8 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐶 ↑o suc 𝑦) ∈ On)
55		ontr1 6382	. . . . . . . 8 ⊢ ((𝐶 ↑o suc 𝑦) ∈ On → (((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦) ∧ (𝐶 ↑o 𝑦) ∈ (𝐶 ↑o suc 𝑦)) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝑦)))
56	54, 55	syl 17	. . . . . . 7 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦) ∧ (𝐶 ↑o 𝑦) ∈ (𝐶 ↑o suc 𝑦)) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝑦)))
57	51, 56	mpan2d 694	. . . . . 6 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝑦)))
58	57	expcom 413	. . . . 5 ⊢ (𝑦 ∈ On → (𝐶 ∈ (On ∖ 2o) → ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝑦))))
59	58	adantr 480	. . . 4 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ 𝑦) → (𝐶 ∈ (On ∖ 2o) → ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝑦))))
60	59	a2d 29	. . 3 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ 𝑦) → ((𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦)) → (𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝑦))))
61		bi2.04 387	. . . . . 6 ⊢ ((𝐴 ∈ 𝑦 → (𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦))) ↔ (𝐶 ∈ (On ∖ 2o) → (𝐴 ∈ 𝑦 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦))))
62	61	ralbii 3076	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦))) ↔ ∀𝑦 ∈ 𝑥 (𝐶 ∈ (On ∖ 2o) → (𝐴 ∈ 𝑦 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦))))
63		r19.21v 3159	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝐶 ∈ (On ∖ 2o) → (𝐴 ∈ 𝑦 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦))) ↔ (𝐶 ∈ (On ∖ 2o) → ∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦))))
64	62, 63	bitri 275	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦))) ↔ (𝐶 ∈ (On ∖ 2o) → ∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦))))
65		limsuc 7828	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥))
66	65	biimpa 476	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → suc 𝐴 ∈ 𝑥)
67		elex 3471	. . . . . . . . . . . . 13 ⊢ (suc 𝐴 ∈ 𝑥 → suc 𝐴 ∈ V)
68		sucexb 7783	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
69		sucidg 6418	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
70	68, 69	sylbir 235	. . . . . . . . . . . . 13 ⊢ (suc 𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
71	67, 70	syl 17	. . . . . . . . . . . 12 ⊢ (suc 𝐴 ∈ 𝑥 → 𝐴 ∈ suc 𝐴)
72		eleq2 2818	. . . . . . . . . . . . . 14 ⊢ (𝑦 = suc 𝐴 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ suc 𝐴))
73		oveq2 7398	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = suc 𝐴 → (𝐶 ↑o 𝑦) = (𝐶 ↑o suc 𝐴))
74	73	eleq2d 2815	. . . . . . . . . . . . . 14 ⊢ (𝑦 = suc 𝐴 → ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦) ↔ (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝐴)))
75	72, 74	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑦 = suc 𝐴 → ((𝐴 ∈ 𝑦 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦)) ↔ (𝐴 ∈ suc 𝐴 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝐴))))
76	75	rspcv 3587	. . . . . . . . . . . 12 ⊢ (suc 𝐴 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦)) → (𝐴 ∈ suc 𝐴 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝐴))))
77	71, 76	mpid 44	. . . . . . . . . . 11 ⊢ (suc 𝐴 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦)) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝐴)))
78	77	anc2li 555	. . . . . . . . . 10 ⊢ (suc 𝐴 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦)) → (suc 𝐴 ∈ 𝑥 ∧ (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝐴))))
79	73	eliuni 4964	. . . . . . . . . 10 ⊢ ((suc 𝐴 ∈ 𝑥 ∧ (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o suc 𝐴)) → (𝐶 ↑o 𝐴) ∈ ∪ 𝑦 ∈ 𝑥 (𝐶 ↑o 𝑦))
80	78, 79	syl6 35	. . . . . . . . 9 ⊢ (suc 𝐴 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦)) → (𝐶 ↑o 𝐴) ∈ ∪ 𝑦 ∈ 𝑥 (𝐶 ↑o 𝑦)))
81	66, 80	syl 17	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦)) → (𝐶 ↑o 𝐴) ∈ ∪ 𝑦 ∈ 𝑥 (𝐶 ↑o 𝑦)))
82	81	adantr 480	. . . . . . 7 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) ∧ 𝐶 ∈ (On ∖ 2o)) → (∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦)) → (𝐶 ↑o 𝐴) ∈ ∪ 𝑦 ∈ 𝑥 (𝐶 ↑o 𝑦)))
83	13	adantl 481	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐶 ∈ On)
84		simpl 482	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐶 ∈ (On ∖ 2o)) → Lim 𝑥)
85	23	adantl 481	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐶 ∈ (On ∖ 2o)) → ∅ ∈ 𝐶)
86		vex 3454	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
87		oelim 8501	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐶) → (𝐶 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ↑o 𝑦))
88	86, 87	mpanlr1 706	. . . . . . . . . 10 ⊢ (((𝐶 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐶) → (𝐶 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ↑o 𝑦))
89	83, 84, 85, 88	syl21anc 837	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ↑o 𝑦))
90	89	adantlr 715	. . . . . . . 8 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ↑o 𝑦))
91	90	eleq2d 2815	. . . . . . 7 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑥) ↔ (𝐶 ↑o 𝐴) ∈ ∪ 𝑦 ∈ 𝑥 (𝐶 ↑o 𝑦)))
92	82, 91	sylibrd 259	. . . . . 6 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) ∧ 𝐶 ∈ (On ∖ 2o)) → (∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦)) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑥)))
93	92	ex 412	. . . . 5 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → (𝐶 ∈ (On ∖ 2o) → (∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦)) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑥))))
94	93	a2d 29	. . . 4 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → ((𝐶 ∈ (On ∖ 2o) → ∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦))) → (𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑥))))
95	64, 94	biimtrid 242	. . 3 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑦))) → (𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝑥))))
96	3, 6, 9, 12, 34, 60, 95	tfindsg2 7841	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐶 ∈ (On ∖ 2o) → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))
97	96	impancom 451	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   ∖ cdif 3914  ∅c0 4299  ∪ ciun 4958  Oncon0 6335  Lim wlim 6336  suc csuc 6337  (class class class)co 7390  1_oc1o 8430  2_oc2o 8431   ·_o comu 8435   ↑_o coe 8436

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-omul 8442  df-oexp 8443

This theorem is referenced by:  oeord  8555  oecan  8556  oeworde  8560  oelimcl  8567  oeord2lim  43305  oeord2i  43306  omcl2  43329