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Theorem oelim2 8505

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 6366	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		0ellim 6365	. . . . . . 7 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
3	2	adantl 481	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∅ ∈ 𝐵)
4		oe0m1 8431	. . . . . . 7 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑_o 𝐵) = ∅))
5	4	biimpa 476	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑_o 𝐵) = ∅)
6	1, 3, 5	syl2anc 584	. . . . 5 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (∅ ↑_o 𝐵) = ∅)
7		eldif 3907	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ 1_o) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 1_o))
8		limord 6362	. . . . . . . . . . . 12 ⊢ (Lim 𝐵 → Ord 𝐵)
9		ordelon 6325	. . . . . . . . . . . 12 ⊢ ((Ord 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
10	8, 9	sylan 580	. . . . . . . . . . 11 ⊢ ((Lim 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
11		on0eln0 6358	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ 𝑥 ≠ ∅))
12		el1o 8405	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 1_o ↔ 𝑥 = ∅)
13	12	necon3bbii 2975	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 ∈ 1_o ↔ 𝑥 ≠ ∅)
14	11, 13	bitr4di 289	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ ¬ 𝑥 ∈ 1_o))
15		oe0m1 8431	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑_o 𝑥) = ∅))
16	15	biimpd 229	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 → (∅ ↑_o 𝑥) = ∅))
17	14, 16	sylbird 260	. . . . . . . . . . 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 454	. . . . . . . . 9 ⊢ ((Lim 𝐵 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 1_o)) → (∅ ↑_o 𝑥) = ∅)
20	7, 19	sylan2b 594	. . . . . . . 8 ⊢ ((Lim 𝐵 ∧ 𝑥 ∈ (𝐵 ∖ 1_o)) → (∅ ↑_o 𝑥) = ∅)
21	20	iuneq2dv 4961	. . . . . . 7 ⊢ (Lim 𝐵 → ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(∅ ↑_o 𝑥) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)∅)
22		df-1o 8380	. . . . . . . . . 10 ⊢ 1_o = suc ∅
23		limsuc 7774	. . . . . . . . . . 11 ⊢ (Lim 𝐵 → (∅ ∈ 𝐵 ↔ suc ∅ ∈ 𝐵))
24	2, 23	mpbid 232	. . . . . . . . . 10 ⊢ (Lim 𝐵 → suc ∅ ∈ 𝐵)
25	22, 24	eqeltrid 2835	. . . . . . . . 9 ⊢ (Lim 𝐵 → 1_o ∈ 𝐵)
26		1on 8392	. . . . . . . . . 10 ⊢ 1_o ∈ On
27	26	onirri 6415	. . . . . . . . 9 ⊢ ¬ 1_o ∈ 1_o
28		eldif 3907	. . . . . . . . 9 ⊢ (1_o ∈ (𝐵 ∖ 1_o) ↔ (1_o ∈ 𝐵 ∧ ¬ 1_o ∈ 1_o))
29	25, 27, 28	sylanblrc 590	. . . . . . . 8 ⊢ (Lim 𝐵 → 1_o ∈ (𝐵 ∖ 1_o))
30		ne0i 4286	. . . . . . . 8 ⊢ (1_o ∈ (𝐵 ∖ 1_o) → (𝐵 ∖ 1_o) ≠ ∅)
31		iunconst 4946	. . . . . . . 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 481	. . . . 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 7348	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑_o 𝐵) = (∅ ↑_o 𝐵))
37		oveq1 7348	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ↑_o 𝑥) = (∅ ↑_o 𝑥))
38	37	iuneq2d 4967	. . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(∅ ↑_o 𝑥))
39	36, 38	eqeq12d 2747	. . . 4 ⊢ (𝐴 = ∅ → ((𝐴 ↑_o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥) ↔ (∅ ↑_o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(∅ ↑_o 𝑥)))
40	35, 39	imbitrrid 246	. . 3 ⊢ (𝐴 = ∅ → ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (𝐴 ↑_o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥)))
41	40	impcom 407	. 2 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝐴 = ∅) → (𝐴 ↑_o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥))
42		oelim 8444	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑_o 𝐵) = ∪ 𝑦 ∈ 𝐵 (𝐴 ↑_o 𝑦))
43		limsuc 7774	. . . . . . . . . . . . 13 ⊢ (Lim 𝐵 → (𝑦 ∈ 𝐵 ↔ suc 𝑦 ∈ 𝐵))
44	43	biimpa 476	. . . . . . . . . . . 12 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → suc 𝑦 ∈ 𝐵)
45		nsuceq0 6386	. . . . . . . . . . . 12 ⊢ suc 𝑦 ≠ ∅
46		dif1o 8410	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ∈ (𝐵 ∖ 1_o) ↔ (suc 𝑦 ∈ 𝐵 ∧ suc 𝑦 ≠ ∅))
47	44, 45, 46	sylanblrc 590	. . . . . . . . . . 11 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → suc 𝑦 ∈ (𝐵 ∖ 1_o))
48	47	ex 412	. . . . . . . . . 10 ⊢ (Lim 𝐵 → (𝑦 ∈ 𝐵 → suc 𝑦 ∈ (𝐵 ∖ 1_o)))
49	48	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → suc 𝑦 ∈ (𝐵 ∖ 1_o)))
50		sssucid 6383	. . . . . . . . . . 11 ⊢ 𝑦 ⊆ suc 𝑦
51		ordelon 6325	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
52	8, 51	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
53		onsuc 7738	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
54	52, 53	jccir 521	. . . . . . . . . . . . . . 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 1118	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ On ∧ suc 𝑦 ∈ On) ∧ 𝐴 ∈ On) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
57	56	ancoms 458	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ (𝑦 ∈ On ∧ suc 𝑦 ∈ On)) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
58	54, 57	sylan2 593	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ (Lim 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
59	58	anassrs 467	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
60		oewordi 8501	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ⊆ suc 𝑦 → (𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o suc 𝑦)))
61	59, 60	sylan 580	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ⊆ suc 𝑦 → (𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o suc 𝑦)))
62	61	an32s 652	. . . . . . . . . . 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 412	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → (𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o suc 𝑦)))
65	49, 64	jcad 512	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → (suc 𝑦 ∈ (𝐵 ∖ 1_o) ∧ (𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o suc 𝑦))))
66		oveq2 7349	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑_o 𝑥) = (𝐴 ↑_o suc 𝑦))
67	66	sseq2d 3962	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o 𝑥) ↔ (𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o suc 𝑦)))
68	67	rspcev 3572	. . . . . . . 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 3123	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑦) ⊆ (𝐴 ↑_o 𝑥))
71		iunss2 4993	. . . . . 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 4081	. . . . . . . 8 ⊢ (𝐵 ∖ 1_o) ⊆ 𝐵
74		iunss1 4951	. . . . . . . 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 7349	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 ↑_o 𝑥) = (𝐴 ↑_o 𝑦))
77	76	cbviunv 4984	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ↑_o 𝑥) = ∪ 𝑦 ∈ 𝐵 (𝐴 ↑_o 𝑦)
78	75, 77	sseqtri 3978	. . . . . 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 3947	. . . 4 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∪ 𝑦 ∈ 𝐵 (𝐴 ↑_o 𝑦) = ∪ 𝑥 ∈ (𝐵 ∖ 1_o)(𝐴 ↑_o 𝑥))
81	80	adantlrl 720	. . 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 8423	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 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 ∖ cdif 3894 ⊆ wss 3897 ∅c0 4278 ∪ ciun 4936 Ord word 6300 Oncon0 6301 Lim wlim 6302 suc csuc 6303 (class class class)co 7341 1_oc1o 8373 ↑_o coe 8379

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pr 5365 ax-un 7663

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-omul 8385 df-oexp 8386

This theorem is referenced by: oelimcl 8510 oaabs2 8559 omabs 8561

W3C validator