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Theorem oelim2 8625

Description: Ordinal exponentiation with a limit exponent. Part of Exercise 4.36 of [Mendelson] p. 250. (Contributed by NM, 6-Jan-2005.)

**Assertion**
Ref	Expression
oelim2	⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑_o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 Allowed substitution hint: 𝐶(𝑥)

Proof of Theorem oelim2 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limelon 6440	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		0ellim 6439	. . . . . . 7 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
3	2	adantl 480	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∅ ∈ 𝐵)
4		oe0m1 8551	. . . . . . 7 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑_o 𝐵) = ∅))
5	4	biimpa 475	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑_o 𝐵) = ∅)
6	1, 3, 5	syl2anc 582	. . . . 5 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (∅ ↑_o 𝐵) = ∅)
7		eldif 3957	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ 1_o) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 1_o))
8		limord 6436	. . . . . . . . . . . 12 ⊢ (Lim 𝐵 → Ord 𝐵)
9		ordelon 6400	. . . . . . . . . . . 12 ⊢ ((Ord 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
10	8, 9	sylan 578	. . . . . . . . . . 11 ⊢ ((Lim 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
11		on0eln0 6432	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ 𝑥 ≠ ∅))
12		el1o 8525	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 1_o ↔ 𝑥 = ∅)
13	12	necon3bbii 2978	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 ∈ 1_o ↔ 𝑥 ≠ ∅)
14	11, 13	bitr4di 288	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ ¬ 𝑥 ∈ 1_o))
15		oe0m1 8551	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑_o 𝑥) = ∅))
16	15	biimpd 228	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 → (∅ ↑_o 𝑥) = ∅))
17	14, 16	sylbird 259	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (¬ 𝑥 ∈ 1_o → (∅ ↑_o 𝑥) = ∅))
18	10, 17	syl 17	. . . . . . . . . 10 ⊢ ((Lim 𝐵 ∧ 𝑥 ∈ 𝐵) → (¬ 𝑥 ∈ 1_o → (∅ ↑_o 𝑥) = ∅))
19	18	impr 453	. . . . . . . . 9 ⊢ ((Lim 𝐵 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 1_o)) → (∅ ↑_o 𝑥) = ∅)
20	7, 19	sylan2b 592	. . . . . . . 8 ⊢ ((Lim 𝐵 ∧ 𝑥 ∈ (𝐵 ∖ 1_o)) → (∅ ↑_o 𝑥) = ∅)
21	20	iuneq2dv 5025	. . . . . . 7 ⊢ (Lim 𝐵 → ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(∅ ↑_o 𝑥) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)∅)
22		df-1o 8496	. . . . . . . . . 10 ⊢ 1_o = suc ∅
23		limsuc 7859	. . . . . . . . . . 11 ⊢ (Lim 𝐵 → (∅ ∈ 𝐵 ↔ suc ∅ ∈ 𝐵))
24	2, 23	mpbid 231	. . . . . . . . . 10 ⊢ (Lim 𝐵 → suc ∅ ∈ 𝐵)
25	22, 24	eqeltrid 2830	. . . . . . . . 9 ⊢ (Lim 𝐵 → 1_o ∈ 𝐵)
26		1on 8508	. . . . . . . . . 10 ⊢ 1_o ∈ On
27	26	onirri 6489	. . . . . . . . 9 ⊢ ¬ 1_o ∈ 1_o
28		eldif 3957	. . . . . . . . 9 ⊢ (1_o ∈ (𝐵 ∖ 1_o) ↔ (1_o ∈ 𝐵 ∧ ¬ 1_o ∈ 1_o))
29	25, 27, 28	sylanblrc 588	. . . . . . . 8 ⊢ (Lim 𝐵 → 1_o ∈ (𝐵 ∖ 1_o))
30		ne0i 4337	. . . . . . . 8 ⊢ (1_o ∈ (𝐵 ∖ 1_o) → (𝐵 ∖ 1_o) ≠ ∅)
31		iunconst 5010	. . . . . . . 8 ⊢ ((𝐵 ∖ 1_o) ≠ ∅ → ∪ 𝑥 ∈ (𝐵 ∖ 1_o)∅ = ∅)
32	29, 30, 31	3syl 18	. . . . . . 7 ⊢ (Lim 𝐵 → ∪ 𝑥 ∈ (𝐵 ∖ 1_o)∅ = ∅)
33	21, 32	eqtrd 2766	. . . . . 6 ⊢ (Lim 𝐵 → ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(∅ ↑_o 𝑥) = ∅)
34	33	adantl 480	. . . . 5 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(∅ ↑_o 𝑥) = ∅)
35	6, 34	eqtr4d 2769	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (∅ ↑_o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(∅ ↑_o 𝑥))
36		oveq1 7431	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑_o 𝐵) = (∅ ↑_o 𝐵))
37		oveq1 7431	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ↑_o 𝑥) = (∅ ↑_o 𝑥))
38	37	iuneq2d 5030	. . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(∅ ↑_o 𝑥))
39	36, 38	eqeq12d 2742	. . . 4 ⊢ (𝐴 = ∅ → ((𝐴 ↑_o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥) ↔ (∅ ↑_o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(∅ ↑_o 𝑥)))
40	35, 39	imbitrrid 245	. . 3 ⊢ (𝐴 = ∅ → ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (𝐴 ↑_o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥)))
41	40	impcom 406	. 2 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝐴 = ∅) → (𝐴 ↑_o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥))
42		oelim 8564	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑_o 𝐵) = ∪ 𝑦 ∈ 𝐵 (𝐴 ↑_o 𝑦))
43		limsuc 7859	. . . . . . . . . . . . 13 ⊢ (Lim 𝐵 → (𝑦 ∈ 𝐵 ↔ suc 𝑦 ∈ 𝐵))
44	43	biimpa 475	. . . . . . . . . . . 12 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → suc 𝑦 ∈ 𝐵)
45		nsuceq0 6459	. . . . . . . . . . . 12 ⊢ suc 𝑦 ≠ ∅
46		dif1o 8530	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ∈ (𝐵 ∖ 1_o) ↔ (suc 𝑦 ∈ 𝐵 ∧ suc 𝑦 ≠ ∅))
47	44, 45, 46	sylanblrc 588	. . . . . . . . . . 11 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → suc 𝑦 ∈ (𝐵 ∖ 1_o))
48	47	ex 411	. . . . . . . . . 10 ⊢ (Lim 𝐵 → (𝑦 ∈ 𝐵 → suc 𝑦 ∈ (𝐵 ∖ 1_o)))
49	48	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → suc 𝑦 ∈ (𝐵 ∖ 1_o)))
50		sssucid 6456	. . . . . . . . . . 11 ⊢ 𝑦 ⊆ suc 𝑦
51		ordelon 6400	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
52	8, 51	sylan 578	. . . . . . . . . . . . . . . 16 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
53		onsuc 7820	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
54	52, 53	jccir 520	. . . . . . . . . . . . . . 15 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On))
55		id 22	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
56	55	3expa 1115	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ On ∧ suc 𝑦 ∈ On) ∧ 𝐴 ∈ On) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
57	56	ancoms 457	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ (𝑦 ∈ On ∧ suc 𝑦 ∈ On)) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
58	54, 57	sylan2 591	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ (Lim 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
59	58	anassrs 466	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
60		oewordi 8621	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ⊆ suc 𝑦 → (𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o suc 𝑦)))
61	59, 60	sylan 578	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ⊆ suc 𝑦 → (𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o suc 𝑦)))
62	61	an32s 650	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝑦 ⊆ suc 𝑦 → (𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o suc 𝑦)))
63	50, 62	mpi 20	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o suc 𝑦))
64	63	ex 411	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → (𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o suc 𝑦)))
65	49, 64	jcad 511	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → (suc 𝑦 ∈ (𝐵 ∖ 1_o) ∧ (𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o suc 𝑦))))
66		oveq2 7432	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑_o 𝑥) = (𝐴 ↑_o suc 𝑦))
67	66	sseq2d 4012	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o 𝑥) ↔ (𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o suc 𝑦)))
68	67	rspcev 3608	. . . . . . . 8 ⊢ ((suc 𝑦 ∈ (𝐵 ∖ 1_o) ∧ (𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o suc 𝑦)) → ∃𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o 𝑥))
69	65, 68	syl6 35	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o 𝑥)))
70	69	ralrimiv 3135	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o 𝑥))
71		iunss2 5057	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o 𝑥) → ∪ 𝑦 ∈ 𝐵 (𝐴 ↑_o 𝑦) ⊆ ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥))
72	70, 71	syl 17	. . . . 5 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∪ 𝑦 ∈ 𝐵 (𝐴 ↑_o 𝑦) ⊆ ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥))
73		difss 4131	. . . . . . . 8 ⊢ (𝐵 ∖ 1_o) ⊆ 𝐵
74		iunss1 5015	. . . . . . . 8 ⊢ ((𝐵 ∖ 1_o) ⊆ 𝐵 → ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥) ⊆ ∪ 𝑥 ∈ 𝐵 (𝐴 ↑_o 𝑥))
75	73, 74	ax-mp 5	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥) ⊆ ∪ 𝑥 ∈ 𝐵 (𝐴 ↑_o 𝑥)
76		oveq2 7432	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 ↑_o 𝑥) = (𝐴 ↑_o 𝑦))
77	76	cbviunv 5048	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ↑_o 𝑥) = ∪ 𝑦 ∈ 𝐵 (𝐴 ↑_o 𝑦)
78	75, 77	sseqtri 4016	. . . . . 6 ⊢ ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 ↑_o 𝑦)
79	78	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 ↑_o 𝑦))
80	72, 79	eqssd 3997	. . . 4 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∪ 𝑦 ∈ 𝐵 (𝐴 ↑_o 𝑦) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥))
81	80	adantlrl 718	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ∪ 𝑦 ∈ 𝐵 (𝐴 ↑_o 𝑦) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥))
82	42, 81	eqtrd 2766	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑_o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥))
83	41, 82	oe0lem 8543	1 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑_o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4325 ∪ ciun 5001 Ord word 6375 Oncon0 6376 Lim wlim 6377 suc csuc 6378 (class class class)co 7424 1_oc1o 8489 ↑_o coe 8495

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-oadd 8500 df-omul 8501 df-oexp 8502

This theorem is referenced by: oelimcl 8630 oaabs2 8679 omabs 8681

W3C validator