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Theorem leordtval2 23106

Description: The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)

**Hypotheses**
Ref	Expression
leordtval.1	⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
leordtval.2	⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))

**Assertion**
Ref	Expression
leordtval2	⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘(𝐴 ∪ 𝐵)))

Proof of Theorem leordtval2 Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		letsr 18559	. . 3 ⊢ ≤ ∈ TosetRel
2		ledm 18556	. . . 4 ⊢ ℝ^* = dom ≤
3		leordtval.1	. . . . 5 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
4	3	leordtvallem1 23104	. . . 4 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
5		leordtval.2	. . . . 5 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
6	3, 5	leordtvallem2 23105	. . . 4 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
7	2, 4, 6	ordtval 23083	. . 3 ⊢ ( ≤ ∈ TosetRel → (ordTop‘ ≤ ) = (topGen‘(fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵)))))
8	1, 7	ax-mp 5	. 2 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))))
9		snex 5394	. . . . 5 ⊢ {ℝ^*} ∈ V
10		xrex 12953	. . . . . . 7 ⊢ ℝ^* ∈ V
11	10	pwex 5338	. . . . . 6 ⊢ 𝒫 ℝ^* ∈ V
12		eqid 2730	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
13		iocssxr 13399	. . . . . . . . . . . 12 ⊢ (𝑥(,]+∞) ⊆ ℝ^*
14	10, 13	elpwi2 5293	. . . . . . . . . . 11 ⊢ (𝑥(,]+∞) ∈ 𝒫 ℝ^*
15	14	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* → (𝑥(,]+∞) ∈ 𝒫 ℝ^*)
16	12, 15	fmpti 7087	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)):ℝ^⟶𝒫 ℝ^
17		frn 6698	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)):ℝ^⟶𝒫 ℝ^ → ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ⊆ 𝒫 ℝ^*)
18	16, 17	ax-mp 5	. . . . . . . 8 ⊢ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ⊆ 𝒫 ℝ^*
19	3, 18	eqsstri 3996	. . . . . . 7 ⊢ 𝐴 ⊆ 𝒫 ℝ^*
20		eqid 2730	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
21		icossxr 13400	. . . . . . . . . . . 12 ⊢ (-∞[,)𝑥) ⊆ ℝ^*
22	10, 21	elpwi2 5293	. . . . . . . . . . 11 ⊢ (-∞[,)𝑥) ∈ 𝒫 ℝ^*
23	22	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* → (-∞[,)𝑥) ∈ 𝒫 ℝ^*)
24	20, 23	fmpti 7087	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)):ℝ^⟶𝒫 ℝ^
25		frn 6698	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)):ℝ^⟶𝒫 ℝ^ → ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) ⊆ 𝒫 ℝ^*)
26	24, 25	ax-mp 5	. . . . . . . 8 ⊢ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) ⊆ 𝒫 ℝ^*
27	5, 26	eqsstri 3996	. . . . . . 7 ⊢ 𝐵 ⊆ 𝒫 ℝ^*
28	19, 27	unssi 4157	. . . . . 6 ⊢ (𝐴 ∪ 𝐵) ⊆ 𝒫 ℝ^*
29	11, 28	ssexi 5280	. . . . 5 ⊢ (𝐴 ∪ 𝐵) ∈ V
30	9, 29	unex 7723	. . . 4 ⊢ ({ℝ^*} ∪ (𝐴 ∪ 𝐵)) ∈ V
31		ssun2 4145	. . . 4 ⊢ (𝐴 ∪ 𝐵) ⊆ ({ℝ^*} ∪ (𝐴 ∪ 𝐵))
32		fiss 9382	. . . 4 ⊢ ((({ℝ^} ∪ (𝐴 ∪ 𝐵)) ∈ V ∧ (𝐴 ∪ 𝐵) ⊆ ({ℝ^} ∪ (𝐴 ∪ 𝐵))) → (fi‘(𝐴 ∪ 𝐵)) ⊆ (fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))))
33	30, 31, 32	mp2an 692	. . 3 ⊢ (fi‘(𝐴 ∪ 𝐵)) ⊆ (fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵)))
34		fvex 6874	. . . . 5 ⊢ (topGen‘(fi‘(𝐴 ∪ 𝐵))) ∈ V
35		ovex 7423	. . . . . . . . . 10 ⊢ (0(,]+∞) ∈ V
36		ovex 7423	. . . . . . . . . 10 ⊢ (-∞[,)1) ∈ V
37	35, 36	unipr 4891	. . . . . . . . 9 ⊢ ∪ {(0(,]+∞), (-∞[,)1)} = ((0(,]+∞) ∪ (-∞[,)1))
38		iocssxr 13399	. . . . . . . . . . 11 ⊢ (0(,]+∞) ⊆ ℝ^*
39		icossxr 13400	. . . . . . . . . . 11 ⊢ (-∞[,)1) ⊆ ℝ^*
40	38, 39	unssi 4157	. . . . . . . . . 10 ⊢ ((0(,]+∞) ∪ (-∞[,)1)) ⊆ ℝ^*
41		mnfxr 11238	. . . . . . . . . . . . 13 ⊢ -∞ ∈ ℝ^*
42		0xr 11228	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
43		pnfxr 11235	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
44		mnflt0 13092	. . . . . . . . . . . . . 14 ⊢ -∞ < 0
45		0lepnf 13100	. . . . . . . . . . . . . 14 ⊢ 0 ≤ +∞
46		df-icc 13320	. . . . . . . . . . . . . . 15 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
47		df-ioc 13318	. . . . . . . . . . . . . . 15 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
48		xrltnle 11248	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (0 < 𝑤 ↔ ¬ 𝑤 ≤ 0))
49		xrletr 13125	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 ≤ 0 ∧ 0 ≤ +∞) → 𝑤 ≤ +∞))
50		xrlttr 13107	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < 0 ∧ 0 < 𝑤) → -∞ < 𝑤))
51		xrltle 13116	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (-∞ < 𝑤 → -∞ ≤ 𝑤))
52	51	3adant2 1131	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (-∞ < 𝑤 → -∞ ≤ 𝑤))
53	50, 52	syld 47	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < 0 ∧ 0 < 𝑤) → -∞ ≤ 𝑤))
54	46, 47, 48, 46, 49, 53	ixxun 13329	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < 0 ∧ 0 ≤ +∞)) → ((-∞[,]0) ∪ (0(,]+∞)) = (-∞[,]+∞))
55	44, 45, 54	mpanr12 705	. . . . . . . . . . . . 13 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞[,]0) ∪ (0(,]+∞)) = (-∞[,]+∞))
56	41, 42, 43, 55	mp3an 1463	. . . . . . . . . . . 12 ⊢ ((-∞[,]0) ∪ (0(,]+∞)) = (-∞[,]+∞)
57		1xr 11240	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ^*
58		0lt1 11707	. . . . . . . . . . . . . 14 ⊢ 0 < 1
59		df-ico 13319	. . . . . . . . . . . . . . 15 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
60		xrlelttr 13123	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((𝑤 ≤ 0 ∧ 0 < 1) → 𝑤 < 1))
61	59, 46, 60	ixxss2 13332	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ^* ∧ 0 < 1) → (-∞[,]0) ⊆ (-∞[,)1))
62	57, 58, 61	mp2an 692	. . . . . . . . . . . . 13 ⊢ (-∞[,]0) ⊆ (-∞[,)1)
63		unss1 4151	. . . . . . . . . . . . 13 ⊢ ((-∞[,]0) ⊆ (-∞[,)1) → ((-∞[,]0) ∪ (0(,]+∞)) ⊆ ((-∞[,)1) ∪ (0(,]+∞)))
64	62, 63	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((-∞[,]0) ∪ (0(,]+∞)) ⊆ ((-∞[,)1) ∪ (0(,]+∞))
65	56, 64	eqsstrri 3997	. . . . . . . . . . 11 ⊢ (-∞[,]+∞) ⊆ ((-∞[,)1) ∪ (0(,]+∞))
66		iccmax 13391	. . . . . . . . . . 11 ⊢ (-∞[,]+∞) = ℝ^*
67		uncom 4124	. . . . . . . . . . 11 ⊢ ((-∞[,)1) ∪ (0(,]+∞)) = ((0(,]+∞) ∪ (-∞[,)1))
68	65, 66, 67	3sstr3i 4000	. . . . . . . . . 10 ⊢ ℝ^* ⊆ ((0(,]+∞) ∪ (-∞[,)1))
69	40, 68	eqssi 3966	. . . . . . . . 9 ⊢ ((0(,]+∞) ∪ (-∞[,)1)) = ℝ^*
70	37, 69	eqtri 2753	. . . . . . . 8 ⊢ ∪ {(0(,]+∞), (-∞[,)1)} = ℝ^*
71		fvex 6874	. . . . . . . . 9 ⊢ (fi‘(𝐴 ∪ 𝐵)) ∈ V
72		ssun1 4144	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
73		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (0(,]+∞) = (0(,]+∞)
74		oveq1 7397	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 0 → (𝑥(,]+∞) = (0(,]+∞))
75	74	rspceeqv 3614	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ (0(,]+∞) = (0(,]+∞)) → ∃𝑥 ∈ ℝ^* (0(,]+∞) = (𝑥(,]+∞))
76	42, 73, 75	mp2an 692	. . . . . . . . . . . . . 14 ⊢ ∃𝑥 ∈ ℝ^* (0(,]+∞) = (𝑥(,]+∞)
77		ovex 7423	. . . . . . . . . . . . . . 15 ⊢ (𝑥(,]+∞) ∈ V
78	12, 77	elrnmpti 5929	. . . . . . . . . . . . . 14 ⊢ ((0(,]+∞) ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ↔ ∃𝑥 ∈ ℝ^* (0(,]+∞) = (𝑥(,]+∞))
79	76, 78	mpbir 231	. . . . . . . . . . . . 13 ⊢ (0(,]+∞) ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
80	79, 3	eleqtrri 2828	. . . . . . . . . . . 12 ⊢ (0(,]+∞) ∈ 𝐴
81	72, 80	sselii 3946	. . . . . . . . . . 11 ⊢ (0(,]+∞) ∈ (𝐴 ∪ 𝐵)
82		ssun2 4145	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
83		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (-∞[,)1) = (-∞[,)1)
84		oveq2 7398	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (-∞[,)𝑥) = (-∞[,)1))
85	84	rspceeqv 3614	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ^* ∧ (-∞[,)1) = (-∞[,)1)) → ∃𝑥 ∈ ℝ^* (-∞[,)1) = (-∞[,)𝑥))
86	57, 83, 85	mp2an 692	. . . . . . . . . . . . . 14 ⊢ ∃𝑥 ∈ ℝ^* (-∞[,)1) = (-∞[,)𝑥)
87		ovex 7423	. . . . . . . . . . . . . . 15 ⊢ (-∞[,)𝑥) ∈ V
88	20, 87	elrnmpti 5929	. . . . . . . . . . . . . 14 ⊢ ((-∞[,)1) ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) ↔ ∃𝑥 ∈ ℝ^* (-∞[,)1) = (-∞[,)𝑥))
89	86, 88	mpbir 231	. . . . . . . . . . . . 13 ⊢ (-∞[,)1) ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
90	89, 5	eleqtrri 2828	. . . . . . . . . . . 12 ⊢ (-∞[,)1) ∈ 𝐵
91	82, 90	sselii 3946	. . . . . . . . . . 11 ⊢ (-∞[,)1) ∈ (𝐴 ∪ 𝐵)
92		prssi 4788	. . . . . . . . . . 11 ⊢ (((0(,]+∞) ∈ (𝐴 ∪ 𝐵) ∧ (-∞[,)1) ∈ (𝐴 ∪ 𝐵)) → {(0(,]+∞), (-∞[,)1)} ⊆ (𝐴 ∪ 𝐵))
93	81, 91, 92	mp2an 692	. . . . . . . . . 10 ⊢ {(0(,]+∞), (-∞[,)1)} ⊆ (𝐴 ∪ 𝐵)
94		ssfii 9377	. . . . . . . . . . 11 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∪ 𝐵) ⊆ (fi‘(𝐴 ∪ 𝐵)))
95	29, 94	ax-mp 5	. . . . . . . . . 10 ⊢ (𝐴 ∪ 𝐵) ⊆ (fi‘(𝐴 ∪ 𝐵))
96	93, 95	sstri 3959	. . . . . . . . 9 ⊢ {(0(,]+∞), (-∞[,)1)} ⊆ (fi‘(𝐴 ∪ 𝐵))
97		eltg3i 22855	. . . . . . . . 9 ⊢ (((fi‘(𝐴 ∪ 𝐵)) ∈ V ∧ {(0(,]+∞), (-∞[,)1)} ⊆ (fi‘(𝐴 ∪ 𝐵))) → ∪ {(0(,]+∞), (-∞[,)1)} ∈ (topGen‘(fi‘(𝐴 ∪ 𝐵))))
98	71, 96, 97	mp2an 692	. . . . . . . 8 ⊢ ∪ {(0(,]+∞), (-∞[,)1)} ∈ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
99	70, 98	eqeltrri 2826	. . . . . . 7 ⊢ ℝ^* ∈ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
100		snssi 4775	. . . . . . 7 ⊢ (ℝ^* ∈ (topGen‘(fi‘(𝐴 ∪ 𝐵))) → {ℝ^*} ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵))))
101	99, 100	ax-mp 5	. . . . . 6 ⊢ {ℝ^*} ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
102		bastg 22860	. . . . . . . 8 ⊢ ((fi‘(𝐴 ∪ 𝐵)) ∈ V → (fi‘(𝐴 ∪ 𝐵)) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵))))
103	71, 102	ax-mp 5	. . . . . . 7 ⊢ (fi‘(𝐴 ∪ 𝐵)) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
104	95, 103	sstri 3959	. . . . . 6 ⊢ (𝐴 ∪ 𝐵) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
105	101, 104	unssi 4157	. . . . 5 ⊢ ({ℝ^*} ∪ (𝐴 ∪ 𝐵)) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
106		fiss 9382	. . . . 5 ⊢ (((topGen‘(fi‘(𝐴 ∪ 𝐵))) ∈ V ∧ ({ℝ^} ∪ (𝐴 ∪ 𝐵)) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))) → (fi‘({ℝ^} ∪ (𝐴 ∪ 𝐵))) ⊆ (fi‘(topGen‘(fi‘(𝐴 ∪ 𝐵)))))
107	34, 105, 106	mp2an 692	. . . 4 ⊢ (fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))) ⊆ (fi‘(topGen‘(fi‘(𝐴 ∪ 𝐵))))
108		fibas 22871	. . . . 5 ⊢ (fi‘(𝐴 ∪ 𝐵)) ∈ TopBases
109		tgcl 22863	. . . . 5 ⊢ ((fi‘(𝐴 ∪ 𝐵)) ∈ TopBases → (topGen‘(fi‘(𝐴 ∪ 𝐵))) ∈ Top)
110		fitop 22794	. . . . 5 ⊢ ((topGen‘(fi‘(𝐴 ∪ 𝐵))) ∈ Top → (fi‘(topGen‘(fi‘(𝐴 ∪ 𝐵)))) = (topGen‘(fi‘(𝐴 ∪ 𝐵))))
111	108, 109, 110	mp2b 10	. . . 4 ⊢ (fi‘(topGen‘(fi‘(𝐴 ∪ 𝐵)))) = (topGen‘(fi‘(𝐴 ∪ 𝐵)))
112	107, 111	sseqtri 3998	. . 3 ⊢ (fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
113		2basgen 22884	. . 3 ⊢ (((fi‘(𝐴 ∪ 𝐵)) ⊆ (fi‘({ℝ^} ∪ (𝐴 ∪ 𝐵))) ∧ (fi‘({ℝ^} ∪ (𝐴 ∪ 𝐵))) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))) → (topGen‘(fi‘(𝐴 ∪ 𝐵))) = (topGen‘(fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵)))))
114	33, 112, 113	mp2an 692	. 2 ⊢ (topGen‘(fi‘(𝐴 ∪ 𝐵))) = (topGen‘(fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))))
115	8, 114	eqtr4i 2756	1 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘(𝐴 ∪ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∃wrex 3054 Vcvv 3450 ∪ cun 3915 ⊆ wss 3917 𝒫 cpw 4566 {csn 4592 {cpr 4594 ∪ cuni 4874 class class class wbr 5110 ↦ cmpt 5191 ran crn 5642 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ficfi 9368 0cc0 11075 1c1 11076 +∞cpnf 11212 -∞cmnf 11213 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 (,]cioc 13314 [,)cico 13315 [,]cicc 13316 topGenctg 17407 ordTopcordt 17469 TosetRel ctsr 18531 Topctop 22787 TopBasesctb 22839

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-1o 8437 df-2o 8438 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fi 9369 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-ioc 13318 df-ico 13319 df-icc 13320 df-topgen 17413 df-ordt 17471 df-ps 18532 df-tsr 18533 df-top 22788 df-bases 22840

This theorem is referenced by: leordtval 23107 lecldbas 23113

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