Theorem oewordri 8601

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 7428	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
2		oveq2 7428	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ↑o 𝑥) = (𝐵 ↑o ∅))
3	1, 2	sseq12d 3941	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (𝐴 ↑o ∅) ⊆ (𝐵 ↑o ∅)))
4		oveq2 7428	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
5		oveq2 7428	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ↑o 𝑥) = (𝐵 ↑o 𝑦))
6	4, 5	sseq12d 3941	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦)))
7		oveq2 7428	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
8		oveq2 7428	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ↑o 𝑥) = (𝐵 ↑o suc 𝑦))
9	7, 8	sseq12d 3941	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (𝐴 ↑o suc 𝑦) ⊆ (𝐵 ↑o suc 𝑦)))
10		oveq2 7428	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐶))
11		oveq2 7428	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 ↑o 𝑥) = (𝐵 ↑o 𝐶))
12	10, 11	sseq12d 3941	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))
13		onelon 6387	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
14		oe0 8530	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
15	13, 14	syl 18	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑o ∅) = 1o)
16		oe0 8530	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 ↑o ∅) = 1o)
17	16	adantr 486	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐵 ↑o ∅) = 1o)
18	15, 17	eqtr4d 2798	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑o ∅) = (𝐵 ↑o ∅))
19		eqimss 3966	. . . . 5 ⊢ ((𝐴 ↑o ∅) = (𝐵 ↑o ∅) → (𝐴 ↑o ∅) ⊆ (𝐵 ↑o ∅))
20	18, 19	syl 18	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑o ∅) ⊆ (𝐵 ↑o ∅))
21		simpl 488	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐵 ∈ On)
22		onelss 6405	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
23	22	imp 412	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ 𝐵)
24	13, 21, 23	jca31 524	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵))
25		oecl 8545	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
26	25	3adant2 1149	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
27		oecl 8545	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑o 𝑦) ∈ On)
28	27	3adant1 1148	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑o 𝑦) ∈ On)
29		simp1 1154	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
30		omwordri 8580	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 412	. . . . . . . . . . 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 729	. . . . . . . . . 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 8579	. . . . . . . . . . . . 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 1436	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ⊆ 𝐵 → ((𝐵 ↑o 𝑦) ·o 𝐴) ⊆ ((𝐵 ↑o 𝑦) ·o 𝐵)))
36	35	imp 412	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐵 ↑o 𝑦) ·o 𝐴) ⊆ ((𝐵 ↑o 𝑦) ·o 𝐵))
37	36	adantrr 730	. . . . . . . . . 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 3918	. . . . . . . . 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 8535	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
40	39	3adant2 1149	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
41	40	adantr 486	. . . . . . . . 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 8535	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑o suc 𝑦) = ((𝐵 ↑o 𝑦) ·o 𝐵))
43	42	3adant1 1148	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ↑o suc 𝑦) = ((𝐵 ↑o 𝑦) ·o 𝐵))
44	43	adantr 486	. . . . . . . . 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 3963	. . . . . . . 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 1376	. . . . . . 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 88	. . . . . 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 429	. . . . 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 35	. . . 4 ⊢ (𝑦 ∈ On → ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → (𝐴 ↑o suc 𝑦) ⊆ (𝐵 ↑o suc 𝑦))))
50		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
51		limelon 6428	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
52	50, 51	mpan 703	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
53		0ellim 6427	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
54		oe0m1 8529	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑o 𝑥) = ∅))
55	54	biimpa 482	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ ∅ ∈ 𝑥) → (∅ ↑o 𝑥) = ∅)
56	52, 53, 55	syl2anc 596	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (∅ ↑o 𝑥) = ∅)
57		0ss 4349	. . . . . . . . . 10 ⊢ ∅ ⊆ (𝐵 ↑o 𝑥)
58	56, 57	eqsstrdi 3952	. . . . . . . . 9 ⊢ (Lim 𝑥 → (∅ ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥))
59		oveq1 7427	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝑥) = (∅ ↑o 𝑥))
60	59	sseq1d 3939	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥) ↔ (∅ ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥)))
61	58, 60	imbitrrid 249	. . . . . . . 8 ⊢ (𝐴 = ∅ → (Lim 𝑥 → (𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥)))
62	61	adantl 487	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ 𝐴 = ∅) → (Lim 𝑥 → (𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥)))
63	62	a1dd 51	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ 𝐴 = ∅) → (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → (𝐴 ↑o 𝑥) ⊆ (𝐵 ↑o 𝑥))))
64		ss2iun 4969	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ⊆ (𝐵 ↑o 𝑦) → ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
65		oelim 8542	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
66	50, 65	mpanlr1 719	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
67	66	an32s 665	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
68	67	adantllr 732	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
69	21	anim1i 627	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ Lim 𝑥) → (𝐵 ∈ On ∧ Lim 𝑥))
70		ne0i 4286	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ ∅)
71		on0eln0 6420	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
72	70, 71	imbitrrid 249	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → ∅ ∈ 𝐵))
73	72	imp 412	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → ∅ ∈ 𝐵)
74	73	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ Lim 𝑥) → ∅ ∈ 𝐵)
75		oelim 8542	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → (𝐵 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
76	50, 75	mpanlr1 719	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝐵 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
77	69, 74, 76	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) ∧ Lim 𝑥) → (𝐵 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
78	77	ad4ant24 767	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ↑o 𝑦))
79	68, 78	sseq12d 3941	. . . . . . . 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 249	. . . . . . 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 418	. . . . . 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 8521	. . . . 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 559	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴 ∈ 𝐵)))
84	82, 83	syl11 34	. . . 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 7876	. . 3 ⊢ (𝐶 ∈ On → ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))
86	85	expd 421	. 2 ⊢ (𝐶 ∈ On → (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶))))
87	86	impcom 413	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  Vcvv 3450   ⊆ wss 3895  ∅c0 4278  ∪ ciun 4950  Oncon0 6362  Lim wlim 6363  suc csuc 6364  (class class class)co 7420  1_oc1o 8469   ·_o comu 8474   ↑_o coe 8475

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-oadd 8480  df-omul 8481  df-oexp 8482

This theorem is used by:  oeordsuc  8603  oege2  44309