Theorem oeordi 7667

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 6658	. . . . 5 ⊢ (𝑥 = suc 𝐴 → (𝐶 ↑𝑜 𝑥) = (𝐶 ↑𝑜 suc 𝐴))
2	1	eleq2d 2687	. . . 4 ⊢ (𝑥 = suc 𝐴 → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑥) ↔ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 suc 𝐴)))
3	2	imbi2d 330	. . 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 6658	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐶 ↑𝑜 𝑥) = (𝐶 ↑𝑜 𝑦))
5	4	eleq2d 2687	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑥) ↔ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑦)))
6	5	imbi2d 330	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑥)) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑦))))
7		oveq2 6658	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐶 ↑𝑜 𝑥) = (𝐶 ↑𝑜 suc 𝑦))
8	7	eleq2d 2687	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑥) ↔ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 suc 𝑦)))
9	8	imbi2d 330	. . 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 6658	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐶 ↑𝑜 𝑥) = (𝐶 ↑𝑜 𝐵))
11	10	eleq2d 2687	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑥) ↔ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵)))
12	11	imbi2d 330	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐶 ∈ (On ∖ 2𝑜) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑥)) ↔ (𝐶 ∈ (On ∖ 2𝑜) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵))))
13		eldifi 3732	. . . . . . . 8 ⊢ (𝐶 ∈ (On ∖ 2𝑜) → 𝐶 ∈ On)
14		oecl 7617	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 𝐴) ∈ On)
15	13, 14	sylan 488	. . . . . . 7 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 𝐴) ∈ On)
16		om1 7622	. . . . . . 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 7582	. . . . . . . . 9 ⊢ (𝐶 ∈ (On ∖ 2𝑜) ↔ (𝐶 ∈ On ∧ 1𝑜 ∈ 𝐶))
19	18	simprbi 480	. . . . . . . 8 ⊢ (𝐶 ∈ (On ∖ 2𝑜) → 1𝑜 ∈ 𝐶)
20	19	adantr 481	. . . . . . 7 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 1𝑜 ∈ 𝐶)
21	13	adantr 481	. . . . . . . 8 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 𝐶 ∈ On)
22		simpr 477	. . . . . . . . 9 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 𝐴 ∈ On)
23		dif20el 7585	. . . . . . . . . 10 ⊢ (𝐶 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐶)
24	23	adantr 481	. . . . . . . . 9 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → ∅ ∈ 𝐶)
25		oen0 7666	. . . . . . . . 9 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶 ↑𝑜 𝐴))
26	21, 22, 24, 25	syl21anc 1325	. . . . . . . 8 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → ∅ ∈ (𝐶 ↑𝑜 𝐴))
27		omordi 7646	. . . . . . . 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 1325	. . . . . . 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 2702	. . . . 5 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 𝐴) ∈ ((𝐶 ↑𝑜 𝐴) ·𝑜 𝐶))
31		oesuc 7607	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 suc 𝐴) = ((𝐶 ↑𝑜 𝐴) ·𝑜 𝐶))
32	13, 31	sylan 488	. . . . 5 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 suc 𝐴) = ((𝐶 ↑𝑜 𝐴) ·𝑜 𝐶))
33	30, 32	eleqtrrd 2704	. . . 4 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 suc 𝐴))
34	33	expcom 451	. . 3 ⊢ (𝐴 ∈ On → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 suc 𝐴)))
35		oecl 7617	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ↑𝑜 𝑦) ∈ On)
36	13, 35	sylan 488	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶 ↑𝑜 𝑦) ∈ On)
37		om1 7622	. . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → 1𝑜 ∈ 𝐶)
40	13	adantr 481	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → 𝐶 ∈ On)
41		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → 𝑦 ∈ On)
42	23	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ∅ ∈ 𝐶)
43		oen0 7666	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ 𝐶) → ∅ ∈ (𝐶 ↑𝑜 𝑦))
44	40, 41, 42, 43	syl21anc 1325	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ∅ ∈ (𝐶 ↑𝑜 𝑦))
45		omordi 7646	. . . . . . . . . . 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 1325	. . . . . . . . . 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 2702	. . . . . . . 8 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶 ↑𝑜 𝑦) ∈ ((𝐶 ↑𝑜 𝑦) ·𝑜 𝐶))
49		oesuc 7607	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ↑𝑜 suc 𝑦) = ((𝐶 ↑𝑜 𝑦) ·𝑜 𝐶))
50	13, 49	sylan 488	. . . . . . . 8 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶 ↑𝑜 suc 𝑦) = ((𝐶 ↑𝑜 𝑦) ·𝑜 𝐶))
51	48, 50	eleqtrrd 2704	. . . . . . 7 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶 ↑𝑜 𝑦) ∈ (𝐶 ↑𝑜 suc 𝑦))
52		suceloni 7013	. . . . . . . . 9 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
53		oecl 7617	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ suc 𝑦 ∈ On) → (𝐶 ↑𝑜 suc 𝑦) ∈ On)
54	13, 52, 53	syl2an 494	. . . . . . . 8 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐶 ↑𝑜 suc 𝑦) ∈ On)
55		ontr1 5771	. . . . . . . 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 710	. . . . . 6 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑦) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 suc 𝑦)))
58	57	expcom 451	. . . . 5 ⊢ (𝑦 ∈ On → (𝐶 ∈ (On ∖ 2𝑜) → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑦) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 suc 𝑦))))
59	58	adantr 481	. . . 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		bi2.04 376	. . . . . 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 2980	. . . . 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 2960	. . . . 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 264	. . . 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 7049	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥))
66	65	biimpa 501	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → suc 𝐴 ∈ 𝑥)
67		elex 3212	. . . . . . . . . . . . 13 ⊢ (suc 𝐴 ∈ 𝑥 → suc 𝐴 ∈ V)
68		sucexb 7009	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
69		sucidg 5803	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
70	68, 69	sylbir 225	. . . . . . . . . . . . 13 ⊢ (suc 𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
71	67, 70	syl 17	. . . . . . . . . . . 12 ⊢ (suc 𝐴 ∈ 𝑥 → 𝐴 ∈ suc 𝐴)
72		eleq2 2690	. . . . . . . . . . . . . 14 ⊢ (𝑦 = suc 𝐴 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ suc 𝐴))
73		oveq2 6658	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = suc 𝐴 → (𝐶 ↑𝑜 𝑦) = (𝐶 ↑𝑜 suc 𝐴))
74	73	eleq2d 2687	. . . . . . . . . . . . . 14 ⊢ (𝑦 = suc 𝐴 → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝑦) ↔ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 suc 𝐴)))
75	72, 74	imbi12d 334	. . . . . . . . . . . . 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 3305	. . . . . . . . . . . 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 580	. . . . . . . . . 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 4526	. . . . . . . . . 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 481	. . . . . . 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 482	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐶 ∈ (On ∖ 2𝑜)) → 𝐶 ∈ On)
84		simpl 473	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐶 ∈ (On ∖ 2𝑜)) → Lim 𝑥)
85	23	adantl 482	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ∅ ∈ 𝐶)
86		vex 3203	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
87		oelim 7614	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐶) → (𝐶 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ↑𝑜 𝑦))
88	86, 87	mpanlr1 722	. . . . . . . . . 10 ⊢ (((𝐶 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐶) → (𝐶 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ↑𝑜 𝑦))
89	83, 84, 85, 88	syl21anc 1325	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ↑𝑜 𝑦))
90	89	adantlr 751	. . . . . . . 8 ⊢ (((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ↑𝑜 𝑦))
91	90	eleq2d 2687	. . . . . . 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 249	. . . . . 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 450	. . . . 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 232	. . 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 7061	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐶 ∈ (On ∖ 2𝑜) → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵)))
97	96	impancom 456	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ∈ 𝐵 → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∖ cdif 3571  ∅c0 3915  ∪ ciun 4520  Oncon0 5723  Lim wlim 5724  suc csuc 5725  (class class class)co 6650  1_𝑜c1o 7553  2_𝑜c2o 7554   ·_𝑜 comu 7558   ↑_𝑜 coe 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by:  oeord  7668  oecan  7669  oeworde  7673  oelimcl  7680