Theorem oeordi 7554

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 6557	. . . . 5 ⊢ (𝑥 = suc 𝐴 → (𝐶 ↑𝑜 𝑥) = (𝐶 ↑𝑜 suc 𝐴))
2	1	eleq2d 2673	. . . 4 ⊢ (𝑥 = suc 𝐴 → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑥) ↔ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 suc 𝐴)))
3	2	imbi2d 329	. . 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 6557	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐶 ↑𝑜 𝑥) = (𝐶 ↑𝑜 𝑦))
5	4	eleq2d 2673	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑥) ↔ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑦)))
6	5	imbi2d 329	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑥)) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑦))))
7		oveq2 6557	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐶 ↑𝑜 𝑥) = (𝐶 ↑𝑜 suc 𝑦))
8	7	eleq2d 2673	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑥) ↔ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 suc 𝑦)))
9	8	imbi2d 329	. . 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 6557	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐶 ↑𝑜 𝑥) = (𝐶 ↑𝑜 𝐵))
11	10	eleq2d 2673	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑥) ↔ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵)))
12	11	imbi2d 329	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑥)) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵))))
13		eldifi 3694	. . . . . . . 8 ⊢ (𝐶 ∈ (On ∖ 2𝑜) → 𝐶 ∈ On)
14		oecl 7504	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 𝐴) ∈ On)
15	13, 14	sylan 487	. . . . . . 7 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 𝐴) ∈ On)
16		om1 7509	. . . . . . 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 7469	. . . . . . . . 9 ⊢ (𝐶 ∈ (On ∖ 2𝑜) ↔ (𝐶 ∈ On ∧ 1𝑜 ∈ 𝐶))
19	18	simprbi 479	. . . . . . . 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 476	. . . . . . . . 9 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 𝐴 ∈ On)
23		dif20el 7472	. . . . . . . . . 10 ⊢ (𝐶 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐶)
24	23	adantr 480	. . . . . . . . 9 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → ∅ ∈ 𝐶)
25		oen0 7553	. . . . . . . . 9 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶 ↑𝑜 𝐴))
26	21, 22, 24, 25	syl21anc 1317	. . . . . . . 8 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → ∅ ∈ (𝐶 ↑𝑜 𝐴))
27		omordi 7533	. . . . . . . 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 1317	. . . . . . 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 2689	. . . . 5 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 𝐴) ∈ ((𝐶 ↑𝑜 𝐴) ·𝑜 𝐶))
31		oesuc 7494	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 suc 𝐴) = ((𝐶 ↑𝑜 𝐴) ·𝑜 𝐶))
32	13, 31	sylan 487	. . . . 5 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 suc 𝐴) = ((𝐶 ↑𝑜 𝐴) ·𝑜 𝐶))
33	30, 32	eleqtrrd 2691	. . . 4 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 suc 𝐴))
34	33	expcom 450	. . 3 ⊢ (𝐴 ∈ On → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 suc 𝐴)))
35		oecl 7504	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ↑𝑜 𝑦) ∈ On)
36	13, 35	sylan 487	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶 ↑𝑜 𝑦) ∈ On)
37		om1 7509	. . . . . . . . . 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 476	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → 𝑦 ∈ On)
42	23	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ∅ ∈ 𝐶)
43		oen0 7553	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶 ↑𝑜 𝑦))
44	40, 41, 42, 43	syl21anc 1317	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ∅ ∈ (𝐶 ↑𝑜 𝑦))
45		omordi 7533	. . . . . . . . . . 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 1317	. . . . . . . . . 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 2689	. . . . . . . 8 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶 ↑𝑜 𝑦) ∈ ((𝐶 ↑𝑜 𝑦) ·𝑜 𝐶))
49		oesuc 7494	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ↑𝑜 suc 𝑦) = ((𝐶 ↑𝑜 𝑦) ·𝑜 𝐶))
50	13, 49	sylan 487	. . . . . . . 8 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶 ↑𝑜 suc 𝑦) = ((𝐶 ↑𝑜 𝑦) ·𝑜 𝐶))
51	48, 50	eleqtrrd 2691	. . . . . . 7 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶 ↑𝑜 𝑦) ∈ (𝐶 ↑𝑜 suc 𝑦))
52		suceloni 6905	. . . . . . . . 9 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
53		oecl 7504	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ suc 𝑦 ∈ On) → (𝐶 ↑𝑜 suc 𝑦) ∈ On)
54	13, 52, 53	syl2an 493	. . . . . . . 8 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶 ↑𝑜 suc 𝑦) ∈ On)
55		ontr1 5688	. . . . . . . 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 706	. . . . . 6 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑦) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 suc 𝑦)))
58	57	expcom 450	. . . . 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		bi2.04 375	. . . . . 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 2963	. . . . 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 2943	. . . . 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 263	. . . 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 6941	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥))
66	65	biimpa 500	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → suc 𝐴 ∈ 𝑥)
67		elex 3185	. . . . . . . . . . . . 13 ⊢ (suc 𝐴 ∈ 𝑥 → suc 𝐴 ∈ V)
68		sucexb 6901	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
69		sucidg 5720	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
70	68, 69	sylbir 224	. . . . . . . . . . . . 13 ⊢ (suc 𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
71	67, 70	syl 17	. . . . . . . . . . . 12 ⊢ (suc 𝐴 ∈ 𝑥 → 𝐴 ∈ suc 𝐴)
72		eleq2 2677	. . . . . . . . . . . . . 14 ⊢ (𝑦 = suc 𝐴 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ suc 𝐴))
73		oveq2 6557	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = suc 𝐴 → (𝐶 ↑𝑜 𝑦) = (𝐶 ↑𝑜 suc 𝐴))
74	73	eleq2d 2673	. . . . . . . . . . . . . 14 ⊢ (𝑦 = suc 𝐴 → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑦) ↔ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 suc 𝐴)))
75	72, 74	imbi12d 333	. . . . . . . . . . . . 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 3278	. . . . . . . . . . . 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 43	. . . . . . . . . . 11 ⊢ (suc 𝐴 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑦)) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 suc 𝐴)))
78	77	anc2li 578	. . . . . . . . . 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 4462	. . . . . . . . . 10 ⊢ ((suc 𝐴 ∈ 𝑥 ∧ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 suc 𝐴)) → (𝐶 ↑𝑜 𝐴) ∈ ∪ 𝑦 ∈ 𝑥 (𝐶 ↑𝑜 𝑦))
80	78, 79	syl6 34	. . . . . . . . 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 472	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐶 ∈ (On ∖ 2𝑜)) → Lim 𝑥)
85	23	adantl 481	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ∅ ∈ 𝐶)
86		vex 3176	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
87		oelim 7501	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐶) → (𝐶 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ↑𝑜 𝑦))
88	86, 87	mpanlr1 718	. . . . . . . . . 10 ⊢ (((𝐶 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐶) → (𝐶 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ↑𝑜 𝑦))
89	83, 84, 85, 88	syl21anc 1317	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ↑𝑜 𝑦))
90	89	adantlr 747	. . . . . . . 8 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ↑𝑜 𝑦))
91	90	eleq2d 2673	. . . . . . 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 248	. . . . . 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 449	. . . . 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	syl5bi 231	. . 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 6953	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵)))
97	96	impancom 455	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ∈ 𝐵 → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  ∪ ciun 4455  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549  1_𝑜c1o 7440  2_𝑜c2o 7441   ·_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-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453

This theorem is referenced by:  oeord  7555  oecan  7556  oeworde  7560  oelimcl  7567