Theorem oelim2 8642

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 ∖ 1o)(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 6454	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		0ellim 6453	. . . . . . 7 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
3	2	adantl 487	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∅ ∈ 𝐵)
4		oe0m1 8567	. . . . . . 7 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
5	4	biimpa 482	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
6	1, 3, 5	syl2anc 596	. . . . 5 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (∅ ↑o 𝐵) = ∅)
7		eldif 4052	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ 1o) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 1o))
8		limord 6450	. . . . . . . . . . . 12 ⊢ (Lim 𝐵 → Ord 𝐵)
9		ordelon 6412	. . . . . . . . . . . 12 ⊢ ((Ord 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
10	8, 9	sylan 592	. . . . . . . . . . 11 ⊢ ((Lim 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
11		on0eln0 6446	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ 𝑥 ≠ ∅))
12		el1o 8541	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅)
13	12	necon3bbii 3006	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 ∈ 1o ↔ 𝑥 ≠ ∅)
14	11, 13	bitr4di 292	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ ¬ 𝑥 ∈ 1o))
15		oe0m1 8567	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑o 𝑥) = ∅))
16	15	biimpd 232	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 → (∅ ↑o 𝑥) = ∅))
17	14, 16	sylbird 263	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (¬ 𝑥 ∈ 1o → (∅ ↑o 𝑥) = ∅))
18	10, 17	syl 18	. . . . . . . . . 10 ⊢ ((Lim 𝐵 ∧ 𝑥 ∈ 𝐵) → (¬ 𝑥 ∈ 1o → (∅ ↑o 𝑥) = ∅))
19	18	impr 460	. . . . . . . . 9 ⊢ ((Lim 𝐵 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 1o)) → (∅ ↑o 𝑥) = ∅)
20	7, 19	sylan2b 606	. . . . . . . 8 ⊢ ((Lim 𝐵 ∧ 𝑥 ∈ (𝐵 ∖ 1o)) → (∅ ↑o 𝑥) = ∅)
21	20	iuneq2dv 4981	. . . . . . 7 ⊢ (Lim 𝐵 → ∪ 𝑥 ∈ (𝐵 ∖ 1o)(∅ ↑o 𝑥) = ∪ 𝑥 ∈ (𝐵 ∖ 1o)∅)
22		df-1o 8514	. . . . . . . . . 10 ⊢ 1o = suc ∅
23		limsuc 7894	. . . . . . . . . . 11 ⊢ (Lim 𝐵 → (∅ ∈ 𝐵 ↔ suc ∅ ∈ 𝐵))
24	2, 23	mpbid 235	. . . . . . . . . 10 ⊢ (Lim 𝐵 → suc ∅ ∈ 𝐵)
25	22, 24	eqeltrid 2868	. . . . . . . . 9 ⊢ (Lim 𝐵 → 1o ∈ 𝐵)
26		1on 8527	. . . . . . . . . 10 ⊢ 1o ∈ On
27	26	onirri 6503	. . . . . . . . 9 ⊢ ¬ 1o ∈ 1o
28		eldif 4052	. . . . . . . . 9 ⊢ (1o ∈ (𝐵 ∖ 1o) ↔ (1o ∈ 𝐵 ∧ ¬ 1o ∈ 1o))
29	25, 27, 28	sylanblrc 602	. . . . . . . 8 ⊢ (Lim 𝐵 → 1o ∈ (𝐵 ∖ 1o))
30		ne0i 4290	. . . . . . . 8 ⊢ (1o ∈ (𝐵 ∖ 1o) → (𝐵 ∖ 1o) ≠ ∅)
31		iunconst 4966	. . . . . . . 8 ⊢ ((𝐵 ∖ 1o) ≠ ∅ → ∪ 𝑥 ∈ (𝐵 ∖ 1o)∅ = ∅)
32	29, 30, 31	3syl 19	. . . . . . 7 ⊢ (Lim 𝐵 → ∪ 𝑥 ∈ (𝐵 ∖ 1o)∅ = ∅)
33	21, 32	eqtrd 2798	. . . . . 6 ⊢ (Lim 𝐵 → ∪ 𝑥 ∈ (𝐵 ∖ 1o)(∅ ↑o 𝑥) = ∅)
34	33	adantl 487	. . . . 5 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∪ 𝑥 ∈ (𝐵 ∖ 1o)(∅ ↑o 𝑥) = ∅)
35	6, 34	eqtr4d 2801	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (∅ ↑o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1o)(∅ ↑o 𝑥))
36		oveq1 7458	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
37		oveq1 7458	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝑥) = (∅ ↑o 𝑥))
38	37	iuneq2d 4986	. . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑥) = ∪ 𝑥 ∈ (𝐵 ∖ 1o)(∅ ↑o 𝑥))
39	36, 38	eqeq12d 2779	. . . 4 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑥) ↔ (∅ ↑o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1o)(∅ ↑o 𝑥)))
40	35, 39	imbitrrid 249	. . 3 ⊢ (𝐴 = ∅ → ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (𝐴 ↑o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑥)))
41	40	impcom 413	. 2 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝐴 = ∅) → (𝐴 ↑o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑥))
42		oelim 8580	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = ∪ 𝑦 ∈ 𝐵 (𝐴 ↑o 𝑦))
43		limsuc 7894	. . . . . . . . . . . . 13 ⊢ (Lim 𝐵 → (𝑦 ∈ 𝐵 ↔ suc 𝑦 ∈ 𝐵))
44	43	biimpa 482	. . . . . . . . . . . 12 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → suc 𝑦 ∈ 𝐵)
45		nsuceq0 6474	. . . . . . . . . . . 12 ⊢ suc 𝑦 ≠ ∅
46		dif1o 8546	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ∈ (𝐵 ∖ 1o) ↔ (suc 𝑦 ∈ 𝐵 ∧ suc 𝑦 ≠ ∅))
47	44, 45, 46	sylanblrc 602	. . . . . . . . . . 11 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → suc 𝑦 ∈ (𝐵 ∖ 1o))
48	47	ex 418	. . . . . . . . . 10 ⊢ (Lim 𝐵 → (𝑦 ∈ 𝐵 → suc 𝑦 ∈ (𝐵 ∖ 1o)))
49	48	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → suc 𝑦 ∈ (𝐵 ∖ 1o)))
50		sssucid 6471	. . . . . . . . . . 11 ⊢ 𝑦 ⊆ suc 𝑦
51		ordelon 6412	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
52	8, 51	sylan 592	. . . . . . . . . . . . . . . 16 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
53		onsuc 7858	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
54	52, 53	jccir 531	. . . . . . . . . . . . . . 15 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On))
55		id 23	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
56	55	3expa 1136	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ On ∧ suc 𝑦 ∈ On) ∧ 𝐴 ∈ On) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
57	56	ancoms 464	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ (𝑦 ∈ On ∧ suc 𝑦 ∈ On)) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
58	54, 57	sylan2 605	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ (Lim 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
59	58	anassrs 473	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
60		oewordi 8638	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ⊆ suc 𝑦 → (𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o suc 𝑦)))
61	59, 60	sylan 592	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ⊆ suc 𝑦 → (𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o suc 𝑦)))
62	61	an32s 665	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝑦 ⊆ suc 𝑦 → (𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o suc 𝑦)))
63	50, 62	mpi 21	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o suc 𝑦))
64	63	ex 418	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → (𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o suc 𝑦)))
65	49, 64	jcad 522	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → (suc 𝑦 ∈ (𝐵 ∖ 1o) ∧ (𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o suc 𝑦))))
66		oveq2 7459	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
67	66	sseq2d 3945	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o 𝑥) ↔ (𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o suc 𝑦)))
68	67	rspcev 3580	. . . . . . . 8 ⊢ ((suc 𝑦 ∈ (𝐵 ∖ 1o) ∧ (𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o suc 𝑦)) → ∃𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o 𝑥))
69	65, 68	syl6 36	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o 𝑥)))
70	69	ralrimiv 3157	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o 𝑥))
71		iunss2 5013	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o 𝑥) → ∪ 𝑦 ∈ 𝐵 (𝐴 ↑o 𝑦) ⊆ ∪ 𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑥))
72	70, 71	syl 18	. . . . 5 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∪ 𝑦 ∈ 𝐵 (𝐴 ↑o 𝑦) ⊆ ∪ 𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑥))
73		difss 4075	. . . . . . . 8 ⊢ (𝐵 ∖ 1o) ⊆ 𝐵
74		iunss1 4971	. . . . . . . 8 ⊢ ((𝐵 ∖ 1o) ⊆ 𝐵 → ∪ 𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑥) ⊆ ∪ 𝑥 ∈ 𝐵 (𝐴 ↑o 𝑥))
75	73, 74	ax-mp 5	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑥) ⊆ ∪ 𝑥 ∈ 𝐵 (𝐴 ↑o 𝑥)
76		oveq2 7459	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
77	76	cbviunv 5002	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝐵 (𝐴 ↑o 𝑦)
78	75, 77	sseqtri 3961	. . . . . 6 ⊢ ∪ 𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 ↑o 𝑦)
79	78	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∪ 𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 ↑o 𝑦))
80	72, 79	eqssd 3930	. . . 4 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∪ 𝑦 ∈ 𝐵 (𝐴 ↑o 𝑦) = ∪ 𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑥))
81	80	adantlrl 733	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ∪ 𝑦 ∈ 𝐵 (𝐴 ↑o 𝑦) = ∪ 𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑥))
82	42, 81	eqtrd 2798	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑥))
83	41, 82	oe0lem 8559	1 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑥))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∖ cdif 4048  ∅c0 4281  ∪ ciun 4956  Ord word 6387  Oncon0 6388  Lim wlim 6389  suc csuc 6390  (class class class)co 7451  1_oc1o 8507   ↑_o coe 8513

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-oadd 8518  df-omul 8519  df-oexp 8520

This theorem is used by:  oelimcl  8647  oaabs2  8696  omabs  8698