Theorem oewordri 8648

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 7456	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
2		oveq2 7456	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ↑o 𝑥) = (𝐵 ↑o ∅))
3	1, 2	sseq12d 4042	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (𝐴 ↑o ∅) ⊆ (𝐵 ↑o ∅)))
4		oveq2 7456	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
5		oveq2 7456	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ↑o 𝑥) = (𝐵 ↑o 𝑦))
6	4, 5	sseq12d 4042	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦)))
7		oveq2 7456	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
8		oveq2 7456	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ↑o 𝑥) = (𝐵 ↑o suc 𝑦))
9	7, 8	sseq12d 4042	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (𝐴 ↑o suc 𝑦) ⊆ (𝐵 ↑o suc 𝑦)))
10		oveq2 7456	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐶))
11		oveq2 7456	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 ↑o 𝑥) = (𝐵 ↑o 𝐶))
12	10, 11	sseq12d 4042	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))
13		onelon 6420	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
14		oe0 8578	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
15	13, 14	syl 17	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑o ∅) = 1o)
16		oe0 8578	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 ↑o ∅) = 1o)
17	16	adantr 480	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐵 ↑o ∅) = 1o)
18	15, 17	eqtr4d 2783	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑o ∅) = (𝐵 ↑o ∅))
19		eqimss 4067	. . . . 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 6437	. . . . . . 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 8593	. . . . . . . . . . . . . 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 8593	. . . . . . . . . . . . . 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 8628	. . . . . . . . . . . . 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 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 715	. . . . . . . . . 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 8627	. . . . . . . . . . . . 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 406	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐵 ↑o 𝑦) ·o 𝐴) ⊆ ((𝐵 ↑o 𝑦) ·o 𝐵))
37	36	adantrr 716	. . . . . . . . . 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 4019	. . . . . . . . 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 8583	. . . . . . . . . . 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 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 8583	. . . . . . . . . . 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 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 4056	. . . . . . . 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 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 3492	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
51		limelon 6459	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
52	50, 51	mpan 689	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
53		0ellim 6458	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
54		oe0m1 8577	. . . . . . . . . . . 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 4423	. . . . . . . . . 10 ⊢ ∅ ⊆ (𝐵 ↑o 𝑥)
58	56, 57	eqsstrdi 4063	. . . . . . . . 9 ⊢ (Lim 𝑥 → (∅ ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥))
59		oveq1 7455	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝑥) = (∅ ↑o 𝑥))
60	59	sseq1d 4040	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (∅ ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥)))
61	58, 60	imbitrrid 246	. . . . . . . 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 5033	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
65		oelim 8590	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
66	50, 65	mpanlr1 705	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
67	66	an32s 651	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
68	67	adantllr 718	. . . . . . . . 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 4364	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ ∅)
71		on0eln0 6451	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
72	70, 71	imbitrrid 246	. . . . . . . . . . . . 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 8590	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → (𝐵 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
76	50, 75	mpanlr1 705	. . . . . . . . . . 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 753	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
79	68, 78	sseq12d 4042	. . . . . . . 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 246	. . . . . . 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 8569	. . . . 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 7902	. . 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 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  ∪ ciun 5015  Oncon0 6395  Lim wlim 6396  suc csuc 6397  (class class class)co 7448  1_oc1o 8515   ·_o comu 8520   ↑_o coe 8521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-omul 8527  df-oexp 8528

This theorem is referenced by:  oeordsuc  8650  oege2  43269