Theorem oewordri 8544

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 7370	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
2		oveq2 7370	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ↑o 𝑥) = (𝐵 ↑o ∅))
3	1, 2	sseq12d 3980	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (𝐴 ↑o ∅) ⊆ (𝐵 ↑o ∅)))
4		oveq2 7370	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
5		oveq2 7370	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ↑o 𝑥) = (𝐵 ↑o 𝑦))
6	4, 5	sseq12d 3980	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦)))
7		oveq2 7370	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
8		oveq2 7370	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ↑o 𝑥) = (𝐵 ↑o suc 𝑦))
9	7, 8	sseq12d 3980	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (𝐴 ↑o suc 𝑦) ⊆ (𝐵 ↑o suc 𝑦)))
10		oveq2 7370	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐶))
11		oveq2 7370	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 ↑o 𝑥) = (𝐵 ↑o 𝐶))
12	10, 11	sseq12d 3980	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))
13		onelon 6347	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
14		oe0 8473	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
15	13, 14	syl 17	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑o ∅) = 1o)
16		oe0 8473	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 ↑o ∅) = 1o)
17	16	adantr 481	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐵 ↑o ∅) = 1o)
18	15, 17	eqtr4d 2774	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑o ∅) = (𝐵 ↑o ∅))
19		eqimss 4005	. . . . 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 483	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐵 ∈ On)
22		onelss 6364	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
23	22	imp 407	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ 𝐵)
24	13, 21, 23	jca31 515	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵))
25		oecl 8488	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
26	25	3adant2 1131	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
27		oecl 8488	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑o 𝑦) ∈ On)
28	27	3adant1 1130	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑o 𝑦) ∈ On)
29		simp1 1136	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
30		omwordri 8524	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 407	. . . . . . . . . . 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 714	. . . . . . . . . 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 8523	. . . . . . . . . . . . 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 1409	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ⊆ 𝐵 → ((𝐵 ↑o 𝑦) ·o 𝐴) ⊆ ((𝐵 ↑o 𝑦) ·o 𝐵)))
36	35	imp 407	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐵 ↑o 𝑦) ·o 𝐴) ⊆ ((𝐵 ↑o 𝑦) ·o 𝐵))
37	36	adantrr 715	. . . . . . . . . 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 3957	. . . . . . . . 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 8478	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
40	39	3adant2 1131	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
41	40	adantr 481	. . . . . . . . 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 8478	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑o suc 𝑦) = ((𝐵 ↑o 𝑦) ·o 𝐵))
43	42	3adant1 1130	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑o suc 𝑦) = ((𝐵 ↑o 𝑦) ·o 𝐵))
44	43	adantr 481	. . . . . . . . 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 3994	. . . . . . . 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 1357	. . . . . . 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 424	. . . . 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 3450	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
51		limelon 6386	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
52	50, 51	mpan 688	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
53		0ellim 6385	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
54		oe0m1 8472	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑o 𝑥) = ∅))
55	54	biimpa 477	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ ∅ ∈ 𝑥) → (∅ ↑o 𝑥) = ∅)
56	52, 53, 55	syl2anc 584	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (∅ ↑o 𝑥) = ∅)
57		0ss 4361	. . . . . . . . . 10 ⊢ ∅ ⊆ (𝐵 ↑o 𝑥)
58	56, 57	eqsstrdi 4001	. . . . . . . . 9 ⊢ (Lim 𝑥 → (∅ ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥))
59		oveq1 7369	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝑥) = (∅ ↑o 𝑥))
60	59	sseq1d 3978	. . . . . . . . 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 482	. . . . . . 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 4977	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
65		oelim 8485	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
66	50, 65	mpanlr1 704	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
67	66	an32s 650	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
68	67	adantllr 717	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
69	21	anim1i 615	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ Lim 𝑥) → (𝐵 ∈ On ∧ Lim 𝑥))
70		ne0i 4299	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ ∅)
71		on0eln0 6378	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
72	70, 71	imbitrrid 245	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → ∅ ∈ 𝐵))
73	72	imp 407	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → ∅ ∈ 𝐵)
74	73	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ Lim 𝑥) → ∅ ∈ 𝐵)
75		oelim 8485	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → (𝐵 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
76	50, 75	mpanlr1 704	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝐵 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
77	69, 74, 76	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ Lim 𝑥) → (𝐵 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
78	77	ad4ant24 752	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
79	68, 78	sseq12d 3980	. . . . . . . 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 413	. . . . . 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 8464	. . . . 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 550	. . . . 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 7806	. . 3 ⊢ (𝐶 ∈ On → ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))
86	85	expd 416	. 2 ⊢ (𝐶 ∈ On → (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶))))
87	86	impcom 408	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  Vcvv 3446   ⊆ wss 3913  ∅c0 4287  ∪ ciun 4959  Oncon0 6322  Lim wlim 6323  suc csuc 6324  (class class class)co 7362  1_oc1o 8410   ·_o comu 8415   ↑_o coe 8416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  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 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-omul 8422  df-oexp 8423

This theorem is referenced by:  oeordsuc  8546  oege2  41700