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Theorem leordtval2 23127

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 18499	. . 3 ⊢ ≤ ∈ TosetRel
2		ledm 18496	. . . 4 ⊢ ℝ^* = dom ≤
3		leordtval.1	. . . . 5 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
4	3	leordtvallem1 23125	. . . 4 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
5		leordtval.2	. . . . 5 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
6	3, 5	leordtvallem2 23126	. . . 4 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
7	2, 4, 6	ordtval 23104	. . 3 ⊢ ( ≤ ∈ TosetRel → (ordTop‘ ≤ ) = (topGen‘(fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵)))))
8	1, 7	ax-mp 5	. 2 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))))
9		snex 5372	. . . . 5 ⊢ {ℝ^*} ∈ V
10		xrex 12885	. . . . . . 7 ⊢ ℝ^* ∈ V
11	10	pwex 5316	. . . . . 6 ⊢ 𝒫 ℝ^* ∈ V
12		eqid 2731	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
13		iocssxr 13331	. . . . . . . . . . . 12 ⊢ (𝑥(,]+∞) ⊆ ℝ^*
14	10, 13	elpwi2 5271	. . . . . . . . . . 11 ⊢ (𝑥(,]+∞) ∈ 𝒫 ℝ^*
15	14	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* → (𝑥(,]+∞) ∈ 𝒫 ℝ^*)
16	12, 15	fmpti 7045	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)):ℝ^⟶𝒫 ℝ^
17		frn 6658	. . . . . . . . 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 3976	. . . . . . 7 ⊢ 𝐴 ⊆ 𝒫 ℝ^*
20		eqid 2731	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
21		icossxr 13332	. . . . . . . . . . . 12 ⊢ (-∞[,)𝑥) ⊆ ℝ^*
22	10, 21	elpwi2 5271	. . . . . . . . . . 11 ⊢ (-∞[,)𝑥) ∈ 𝒫 ℝ^*
23	22	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* → (-∞[,)𝑥) ∈ 𝒫 ℝ^*)
24	20, 23	fmpti 7045	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)):ℝ^⟶𝒫 ℝ^
25		frn 6658	. . . . . . . . 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 3976	. . . . . . 7 ⊢ 𝐵 ⊆ 𝒫 ℝ^*
28	19, 27	unssi 4138	. . . . . 6 ⊢ (𝐴 ∪ 𝐵) ⊆ 𝒫 ℝ^*
29	11, 28	ssexi 5258	. . . . 5 ⊢ (𝐴 ∪ 𝐵) ∈ V
30	9, 29	unex 7677	. . . 4 ⊢ ({ℝ^*} ∪ (𝐴 ∪ 𝐵)) ∈ V
31		ssun2 4126	. . . 4 ⊢ (𝐴 ∪ 𝐵) ⊆ ({ℝ^*} ∪ (𝐴 ∪ 𝐵))
32		fiss 9308	. . . 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 6835	. . . . 5 ⊢ (topGen‘(fi‘(𝐴 ∪ 𝐵))) ∈ V
35		ovex 7379	. . . . . . . . . 10 ⊢ (0(,]+∞) ∈ V
36		ovex 7379	. . . . . . . . . 10 ⊢ (-∞[,)1) ∈ V
37	35, 36	unipr 4873	. . . . . . . . 9 ⊢ ∪ {(0(,]+∞), (-∞[,)1)} = ((0(,]+∞) ∪ (-∞[,)1))
38		iocssxr 13331	. . . . . . . . . . 11 ⊢ (0(,]+∞) ⊆ ℝ^*
39		icossxr 13332	. . . . . . . . . . 11 ⊢ (-∞[,)1) ⊆ ℝ^*
40	38, 39	unssi 4138	. . . . . . . . . 10 ⊢ ((0(,]+∞) ∪ (-∞[,)1)) ⊆ ℝ^*
41		mnfxr 11169	. . . . . . . . . . . . 13 ⊢ -∞ ∈ ℝ^*
42		0xr 11159	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
43		pnfxr 11166	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
44		mnflt0 13024	. . . . . . . . . . . . . 14 ⊢ -∞ < 0
45		0lepnf 13032	. . . . . . . . . . . . . 14 ⊢ 0 ≤ +∞
46		df-icc 13252	. . . . . . . . . . . . . . 15 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
47		df-ioc 13250	. . . . . . . . . . . . . . 15 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
48		xrltnle 11179	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (0 < 𝑤 ↔ ¬ 𝑤 ≤ 0))
49		xrletr 13057	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 ≤ 0 ∧ 0 ≤ +∞) → 𝑤 ≤ +∞))
50		xrlttr 13039	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < 0 ∧ 0 < 𝑤) → -∞ < 𝑤))
51		xrltle 13048	. . . . . . . . . . . . . . . . 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 13261	. . . . . . . . . . . . . 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 11171	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ^*
58		0lt1 11639	. . . . . . . . . . . . . 14 ⊢ 0 < 1
59		df-ico 13251	. . . . . . . . . . . . . . 15 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
60		xrlelttr 13055	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((𝑤 ≤ 0 ∧ 0 < 1) → 𝑤 < 1))
61	59, 46, 60	ixxss2 13264	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ^* ∧ 0 < 1) → (-∞[,]0) ⊆ (-∞[,)1))
62	57, 58, 61	mp2an 692	. . . . . . . . . . . . 13 ⊢ (-∞[,]0) ⊆ (-∞[,)1)
63		unss1 4132	. . . . . . . . . . . . 13 ⊢ ((-∞[,]0) ⊆ (-∞[,)1) → ((-∞[,]0) ∪ (0(,]+∞)) ⊆ ((-∞[,)1) ∪ (0(,]+∞)))
64	62, 63	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((-∞[,]0) ∪ (0(,]+∞)) ⊆ ((-∞[,)1) ∪ (0(,]+∞))
65	56, 64	eqsstrri 3977	. . . . . . . . . . 11 ⊢ (-∞[,]+∞) ⊆ ((-∞[,)1) ∪ (0(,]+∞))
66		iccmax 13323	. . . . . . . . . . 11 ⊢ (-∞[,]+∞) = ℝ^*
67		uncom 4105	. . . . . . . . . . 11 ⊢ ((-∞[,)1) ∪ (0(,]+∞)) = ((0(,]+∞) ∪ (-∞[,)1))
68	65, 66, 67	3sstr3i 3980	. . . . . . . . . 10 ⊢ ℝ^* ⊆ ((0(,]+∞) ∪ (-∞[,)1))
69	40, 68	eqssi 3946	. . . . . . . . 9 ⊢ ((0(,]+∞) ∪ (-∞[,)1)) = ℝ^*
70	37, 69	eqtri 2754	. . . . . . . 8 ⊢ ∪ {(0(,]+∞), (-∞[,)1)} = ℝ^*
71		fvex 6835	. . . . . . . . 9 ⊢ (fi‘(𝐴 ∪ 𝐵)) ∈ V
72		ssun1 4125	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
73		eqid 2731	. . . . . . . . . . . . . . 15 ⊢ (0(,]+∞) = (0(,]+∞)
74		oveq1 7353	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 0 → (𝑥(,]+∞) = (0(,]+∞))
75	74	rspceeqv 3595	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ (0(,]+∞) = (0(,]+∞)) → ∃𝑥 ∈ ℝ^* (0(,]+∞) = (𝑥(,]+∞))
76	42, 73, 75	mp2an 692	. . . . . . . . . . . . . 14 ⊢ ∃𝑥 ∈ ℝ^* (0(,]+∞) = (𝑥(,]+∞)
77		ovex 7379	. . . . . . . . . . . . . . 15 ⊢ (𝑥(,]+∞) ∈ V
78	12, 77	elrnmpti 5901	. . . . . . . . . . . . . 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 2830	. . . . . . . . . . . 12 ⊢ (0(,]+∞) ∈ 𝐴
81	72, 80	sselii 3926	. . . . . . . . . . 11 ⊢ (0(,]+∞) ∈ (𝐴 ∪ 𝐵)
82		ssun2 4126	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
83		eqid 2731	. . . . . . . . . . . . . . 15 ⊢ (-∞[,)1) = (-∞[,)1)
84		oveq2 7354	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (-∞[,)𝑥) = (-∞[,)1))
85	84	rspceeqv 3595	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ^* ∧ (-∞[,)1) = (-∞[,)1)) → ∃𝑥 ∈ ℝ^* (-∞[,)1) = (-∞[,)𝑥))
86	57, 83, 85	mp2an 692	. . . . . . . . . . . . . 14 ⊢ ∃𝑥 ∈ ℝ^* (-∞[,)1) = (-∞[,)𝑥)
87		ovex 7379	. . . . . . . . . . . . . . 15 ⊢ (-∞[,)𝑥) ∈ V
88	20, 87	elrnmpti 5901	. . . . . . . . . . . . . 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 2830	. . . . . . . . . . . 12 ⊢ (-∞[,)1) ∈ 𝐵
91	82, 90	sselii 3926	. . . . . . . . . . 11 ⊢ (-∞[,)1) ∈ (𝐴 ∪ 𝐵)
92		prssi 4770	. . . . . . . . . . 11 ⊢ (((0(,]+∞) ∈ (𝐴 ∪ 𝐵) ∧ (-∞[,)1) ∈ (𝐴 ∪ 𝐵)) → {(0(,]+∞), (-∞[,)1)} ⊆ (𝐴 ∪ 𝐵))
93	81, 91, 92	mp2an 692	. . . . . . . . . 10 ⊢ {(0(,]+∞), (-∞[,)1)} ⊆ (𝐴 ∪ 𝐵)
94		ssfii 9303	. . . . . . . . . . 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 3939	. . . . . . . . 9 ⊢ {(0(,]+∞), (-∞[,)1)} ⊆ (fi‘(𝐴 ∪ 𝐵))
97		eltg3i 22876	. . . . . . . . 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 2828	. . . . . . 7 ⊢ ℝ^* ∈ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
100		snssi 4757	. . . . . . 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 22881	. . . . . . . 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 3939	. . . . . 6 ⊢ (𝐴 ∪ 𝐵) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
105	101, 104	unssi 4138	. . . . 5 ⊢ ({ℝ^*} ∪ (𝐴 ∪ 𝐵)) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
106		fiss 9308	. . . . 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 22892	. . . . 5 ⊢ (fi‘(𝐴 ∪ 𝐵)) ∈ TopBases
109		tgcl 22884	. . . . 5 ⊢ ((fi‘(𝐴 ∪ 𝐵)) ∈ TopBases → (topGen‘(fi‘(𝐴 ∪ 𝐵))) ∈ Top)
110		fitop 22815	. . . . 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 3978	. . 3 ⊢ (fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
113		2basgen 22905	. . 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 2757	1 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘(𝐴 ∪ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∃wrex 3056 Vcvv 3436 ∪ cun 3895 ⊆ wss 3897 𝒫 cpw 4547 {csn 4573 {cpr 4575 ∪ cuni 4856 class class class wbr 5089 ↦ cmpt 5170 ran crn 5615 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ficfi 9294 0cc0 11006 1c1 11007 +∞cpnf 11143 -∞cmnf 11144 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 (,]cioc 13246 [,)cico 13247 [,]cicc 13248 topGenctg 17341 ordTopcordt 17403 TosetRel ctsr 18471 Topctop 22808 TopBasesctb 22860

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 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-1o 8385 df-2o 8386 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fi 9295 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-ioc 13250 df-ico 13251 df-icc 13252 df-topgen 17347 df-ordt 17405 df-ps 18472 df-tsr 18473 df-top 22809 df-bases 22861

This theorem is referenced by: leordtval 23128 lecldbas 23134

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