Theorem oewordri 8596

Description: Weak ordering property of ordinal exponentiation. Proposition 8.35 of [TakeutiZaring] p. 68. (Contributed by NM, 6-Jan-2005.)

Assertion

Ref	Expression
oewordri	⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))

Proof of Theorem oewordri

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7420	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
2		oveq2 7420	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ↑o 𝑥) = (𝐵 ↑o ∅))
3	1, 2	sseq12d 4015	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (𝐴 ↑o ∅) ⊆ (𝐵 ↑o ∅)))
4		oveq2 7420	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
5		oveq2 7420	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ↑o 𝑥) = (𝐵 ↑o 𝑦))
6	4, 5	sseq12d 4015	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦)))
7		oveq2 7420	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
8		oveq2 7420	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ↑o 𝑥) = (𝐵 ↑o suc 𝑦))
9	7, 8	sseq12d 4015	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (𝐴 ↑o suc 𝑦) ⊆ (𝐵 ↑o suc 𝑦)))
10		oveq2 7420	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐶))
11		oveq2 7420	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 ↑o 𝑥) = (𝐵 ↑o 𝐶))
12	10, 11	sseq12d 4015	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))
13		onelon 6389	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
14		oe0 8526	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
15	13, 14	syl 17	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑o ∅) = 1o)
16		oe0 8526	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 ↑o ∅) = 1o)
17	16	adantr 480	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐵 ↑o ∅) = 1o)
18	15, 17	eqtr4d 2774	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑o ∅) = (𝐵 ↑o ∅))
19		eqimss 4040	. . . . 5 ⊢ ((𝐴 ↑o ∅) = (𝐵 ↑o ∅) → (𝐴 ↑o ∅) ⊆ (𝐵 ↑o ∅))
20	18, 19	syl 17	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑o ∅) ⊆ (𝐵 ↑o ∅))
21		simpl 482	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐵 ∈ On)
22		onelss 6406	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
23	22	imp 406	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ 𝐵)
24	13, 21, 23	jca31 514	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵))
25		oecl 8541	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
26	25	3adant2 1130	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
27		oecl 8541	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑o 𝑦) ∈ On)
28	27	3adant1 1129	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑o 𝑦) ∈ On)
29		simp1 1135	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
30		omwordri 8576	. . . . . . . . . . . . 13 ⊢ (((𝐴 ↑o 𝑦) ∈ On ∧ (𝐵 ↑o 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → ((𝐴 ↑o 𝑦) ·o 𝐴) ⊆ ((𝐵 ↑o 𝑦) ·o 𝐴)))
31	26, 28, 29, 30	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → ((𝐴 ↑o 𝑦) ·o 𝐴) ⊆ ((𝐵 ↑o 𝑦) ·o 𝐴)))
32	31	imp 406	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦)) → ((𝐴 ↑o 𝑦) ·o 𝐴) ⊆ ((𝐵 ↑o 𝑦) ·o 𝐴))
33	32	adantrl 713	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦))) → ((𝐴 ↑o 𝑦) ·o 𝐴) ⊆ ((𝐵 ↑o 𝑦) ·o 𝐴))
34		omwordi 8575	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐵 ↑o 𝑦) ∈ On) → (𝐴 ⊆ 𝐵 → ((𝐵 ↑o 𝑦) ·o 𝐴) ⊆ ((𝐵 ↑o 𝑦) ·o 𝐵)))
35	28, 34	syld3an3 1408	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ⊆ 𝐵 → ((𝐵 ↑o 𝑦) ·o 𝐴) ⊆ ((𝐵 ↑o 𝑦) ·o 𝐵)))
36	35	imp 406	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐵 ↑o 𝑦) ·o 𝐴) ⊆ ((𝐵 ↑o 𝑦) ·o 𝐵))
37	36	adantrr 714	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦))) → ((𝐵 ↑o 𝑦) ·o 𝐴) ⊆ ((𝐵 ↑o 𝑦) ·o 𝐵))
38	33, 37	sstrd 3992	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦))) → ((𝐴 ↑o 𝑦) ·o 𝐴) ⊆ ((𝐵 ↑o 𝑦) ·o 𝐵))
39		oesuc 8531	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
40	39	3adant2 1130	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
41	40	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦))) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
42		oesuc 8531	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑o suc 𝑦) = ((𝐵 ↑o 𝑦) ·o 𝐵))
43	42	3adant1 1129	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑o suc 𝑦) = ((𝐵 ↑o 𝑦) ·o 𝐵))
44	43	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦))) → (𝐵 ↑o suc 𝑦) = ((𝐵 ↑o 𝑦) ·o 𝐵))
45	38, 41, 44	3sstr4d 4029	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦))) → (𝐴 ↑o suc 𝑦) ⊆ (𝐵 ↑o suc 𝑦))
46	45	exp520 1356	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (𝐴 ⊆ 𝐵 → ((𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → (𝐴 ↑o suc 𝑦) ⊆ (𝐵 ↑o suc 𝑦))))))
47	46	com3r 87	. . . . . 6 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → (𝐴 ⊆ 𝐵 → ((𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → (𝐴 ↑o suc 𝑦) ⊆ (𝐵 ↑o suc 𝑦))))))
48	47	imp4c 423	. . . . 5 ⊢ (𝑦 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → (𝐴 ↑o suc 𝑦) ⊆ (𝐵 ↑o suc 𝑦))))
49	24, 48	syl5 34	. . . 4 ⊢ (𝑦 ∈ On → ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → (𝐴 ↑o suc 𝑦) ⊆ (𝐵 ↑o suc 𝑦))))
50		vex 3477	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
51		limelon 6428	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
52	50, 51	mpan 687	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
53		0ellim 6427	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
54		oe0m1 8525	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑o 𝑥) = ∅))
55	54	biimpa 476	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ ∅ ∈ 𝑥) → (∅ ↑o 𝑥) = ∅)
56	52, 53, 55	syl2anc 583	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (∅ ↑o 𝑥) = ∅)
57		0ss 4396	. . . . . . . . . 10 ⊢ ∅ ⊆ (𝐵 ↑o 𝑥)
58	56, 57	eqsstrdi 4036	. . . . . . . . 9 ⊢ (Lim 𝑥 → (∅ ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥))
59		oveq1 7419	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝑥) = (∅ ↑o 𝑥))
60	59	sseq1d 4013	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (∅ ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥)))
61	58, 60	imbitrrid 245	. . . . . . . 8 ⊢ (𝐴 = ∅ → (Lim 𝑥 → (𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥)))
62	61	adantl 481	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ 𝐴 = ∅) → (Lim 𝑥 → (𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥)))
63	62	a1dd 50	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ 𝐴 = ∅) → (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → (𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥))))
64		ss2iun 5015	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
65		oelim 8538	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
66	50, 65	mpanlr1 703	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
67	66	an32s 649	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
68	67	adantllr 716	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
69	21	anim1i 614	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ Lim 𝑥) → (𝐵 ∈ On ∧ Lim 𝑥))
70		ne0i 4334	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ ∅)
71		on0eln0 6420	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
72	70, 71	imbitrrid 245	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → ∅ ∈ 𝐵))
73	72	imp 406	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → ∅ ∈ 𝐵)
74	73	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ Lim 𝑥) → ∅ ∈ 𝐵)
75		oelim 8538	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → (𝐵 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
76	50, 75	mpanlr1 703	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝐵 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
77	69, 74, 76	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ Lim 𝑥) → (𝐵 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
78	77	ad4ant24 751	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
79	68, 78	sseq12d 4015	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦)))
80	64, 79	imbitrrid 245	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → (𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥)))
81	80	ex 412	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) → (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → (𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥))))
82	63, 81	oe0lem 8517	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) → (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → (𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥))))
83	13	ancri 549	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)))
84	82, 83	syl11 33	. . . 4 ⊢ (Lim 𝑥 → ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → (𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥))))
85	3, 6, 9, 12, 20, 49, 84	tfinds3 7858	. . 3 ⊢ (𝐶 ∈ On → ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))
86	85	expd 415	. 2 ⊢ (𝐶 ∈ On → (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶))))
87	86	impcom 407	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  Vcvv 3473   ⊆ wss 3948  ∅c0 4322  ∪ ciun 4997  Oncon0 6364  Lim wlim 6365  suc csuc 6366  (class class class)co 7412  1_oc1o 8463   ·_o comu 8468   ↑_o coe 8469

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-oadd 8474  df-omul 8475  df-oexp 8476

This theorem is referenced by:  oeordsuc  8598  oege2  42360