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Theorem leordtval2 22372

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 18320	. . 3 ⊢ ≤ ∈ TosetRel
2		ledm 18317	. . . 4 ⊢ ℝ^* = dom ≤
3		leordtval.1	. . . . 5 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
4	3	leordtvallem1 22370	. . . 4 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
5		leordtval.2	. . . . 5 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
6	3, 5	leordtvallem2 22371	. . . 4 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
7	2, 4, 6	ordtval 22349	. . 3 ⊢ ( ≤ ∈ TosetRel → (ordTop‘ ≤ ) = (topGen‘(fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵)))))
8	1, 7	ax-mp 5	. 2 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))))
9		snex 5355	. . . . 5 ⊢ {ℝ^*} ∈ V
10		xrex 12736	. . . . . . 7 ⊢ ℝ^* ∈ V
11	10	pwex 5304	. . . . . 6 ⊢ 𝒫 ℝ^* ∈ V
12		eqid 2739	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
13		iocssxr 13172	. . . . . . . . . . . 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 6995	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)):ℝ^⟶𝒫 ℝ^
17		frn 6616	. . . . . . . . 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 3956	. . . . . . 7 ⊢ 𝐴 ⊆ 𝒫 ℝ^*
20		eqid 2739	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
21		icossxr 13173	. . . . . . . . . . . 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 6995	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)):ℝ^⟶𝒫 ℝ^
25		frn 6616	. . . . . . . . 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 3956	. . . . . . 7 ⊢ 𝐵 ⊆ 𝒫 ℝ^*
28	19, 27	unssi 4120	. . . . . 6 ⊢ (𝐴 ∪ 𝐵) ⊆ 𝒫 ℝ^*
29	11, 28	ssexi 5247	. . . . 5 ⊢ (𝐴 ∪ 𝐵) ∈ V
30	9, 29	unex 7605	. . . 4 ⊢ ({ℝ^*} ∪ (𝐴 ∪ 𝐵)) ∈ V
31		ssun2 4108	. . . 4 ⊢ (𝐴 ∪ 𝐵) ⊆ ({ℝ^*} ∪ (𝐴 ∪ 𝐵))
32		fiss 9192	. . . 4 ⊢ ((({ℝ^} ∪ (𝐴 ∪ 𝐵)) ∈ V ∧ (𝐴 ∪ 𝐵) ⊆ ({ℝ^} ∪ (𝐴 ∪ 𝐵))) → (fi‘(𝐴 ∪ 𝐵)) ⊆ (fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))))
33	30, 31, 32	mp2an 689	. . 3 ⊢ (fi‘(𝐴 ∪ 𝐵)) ⊆ (fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵)))
34		fvex 6796	. . . . 5 ⊢ (topGen‘(fi‘(𝐴 ∪ 𝐵))) ∈ V
35		ovex 7317	. . . . . . . . . 10 ⊢ (0(,]+∞) ∈ V
36		ovex 7317	. . . . . . . . . 10 ⊢ (-∞[,)1) ∈ V
37	35, 36	unipr 4858	. . . . . . . . 9 ⊢ ∪ {(0(,]+∞), (-∞[,)1)} = ((0(,]+∞) ∪ (-∞[,)1))
38		iocssxr 13172	. . . . . . . . . . 11 ⊢ (0(,]+∞) ⊆ ℝ^*
39		icossxr 13173	. . . . . . . . . . 11 ⊢ (-∞[,)1) ⊆ ℝ^*
40	38, 39	unssi 4120	. . . . . . . . . 10 ⊢ ((0(,]+∞) ∪ (-∞[,)1)) ⊆ ℝ^*
41		mnfxr 11041	. . . . . . . . . . . . 13 ⊢ -∞ ∈ ℝ^*
42		0xr 11031	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
43		pnfxr 11038	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
44		mnflt0 12870	. . . . . . . . . . . . . 14 ⊢ -∞ < 0
45		0lepnf 12877	. . . . . . . . . . . . . 14 ⊢ 0 ≤ +∞
46		df-icc 13095	. . . . . . . . . . . . . . 15 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
47		df-ioc 13093	. . . . . . . . . . . . . . 15 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
48		xrltnle 11051	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (0 < 𝑤 ↔ ¬ 𝑤 ≤ 0))
49		xrletr 12901	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 ≤ 0 ∧ 0 ≤ +∞) → 𝑤 ≤ +∞))
50		xrlttr 12883	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < 0 ∧ 0 < 𝑤) → -∞ < 𝑤))
51		xrltle 12892	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (-∞ < 𝑤 → -∞ ≤ 𝑤))
52	51	3adant2 1130	. . . . . . . . . . . . . . . 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 13104	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < 0 ∧ 0 ≤ +∞)) → ((-∞[,]0) ∪ (0(,]+∞)) = (-∞[,]+∞))
55	44, 45, 54	mpanr12 702	. . . . . . . . . . . . 13 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞[,]0) ∪ (0(,]+∞)) = (-∞[,]+∞))
56	41, 42, 43, 55	mp3an 1460	. . . . . . . . . . . 12 ⊢ ((-∞[,]0) ∪ (0(,]+∞)) = (-∞[,]+∞)
57		1xr 11043	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ^*
58		0lt1 11506	. . . . . . . . . . . . . 14 ⊢ 0 < 1
59		df-ico 13094	. . . . . . . . . . . . . . 15 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
60		xrlelttr 12899	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((𝑤 ≤ 0 ∧ 0 < 1) → 𝑤 < 1))
61	59, 46, 60	ixxss2 13107	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ^* ∧ 0 < 1) → (-∞[,]0) ⊆ (-∞[,)1))
62	57, 58, 61	mp2an 689	. . . . . . . . . . . . 13 ⊢ (-∞[,]0) ⊆ (-∞[,)1)
63		unss1 4114	. . . . . . . . . . . . 13 ⊢ ((-∞[,]0) ⊆ (-∞[,)1) → ((-∞[,]0) ∪ (0(,]+∞)) ⊆ ((-∞[,)1) ∪ (0(,]+∞)))
64	62, 63	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((-∞[,]0) ∪ (0(,]+∞)) ⊆ ((-∞[,)1) ∪ (0(,]+∞))
65	56, 64	eqsstrri 3957	. . . . . . . . . . 11 ⊢ (-∞[,]+∞) ⊆ ((-∞[,)1) ∪ (0(,]+∞))
66		iccmax 13164	. . . . . . . . . . 11 ⊢ (-∞[,]+∞) = ℝ^*
67		uncom 4088	. . . . . . . . . . 11 ⊢ ((-∞[,)1) ∪ (0(,]+∞)) = ((0(,]+∞) ∪ (-∞[,)1))
68	65, 66, 67	3sstr3i 3964	. . . . . . . . . 10 ⊢ ℝ^* ⊆ ((0(,]+∞) ∪ (-∞[,)1))
69	40, 68	eqssi 3938	. . . . . . . . 9 ⊢ ((0(,]+∞) ∪ (-∞[,)1)) = ℝ^*
70	37, 69	eqtri 2767	. . . . . . . 8 ⊢ ∪ {(0(,]+∞), (-∞[,)1)} = ℝ^*
71		fvex 6796	. . . . . . . . 9 ⊢ (fi‘(𝐴 ∪ 𝐵)) ∈ V
72		ssun1 4107	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
73		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (0(,]+∞) = (0(,]+∞)
74		oveq1 7291	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 0 → (𝑥(,]+∞) = (0(,]+∞))
75	74	rspceeqv 3576	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ (0(,]+∞) = (0(,]+∞)) → ∃𝑥 ∈ ℝ^* (0(,]+∞) = (𝑥(,]+∞))
76	42, 73, 75	mp2an 689	. . . . . . . . . . . . . 14 ⊢ ∃𝑥 ∈ ℝ^* (0(,]+∞) = (𝑥(,]+∞)
77		ovex 7317	. . . . . . . . . . . . . . 15 ⊢ (𝑥(,]+∞) ∈ V
78	12, 77	elrnmpti 5872	. . . . . . . . . . . . . 14 ⊢ ((0(,]+∞) ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ↔ ∃𝑥 ∈ ℝ^* (0(,]+∞) = (𝑥(,]+∞))
79	76, 78	mpbir 230	. . . . . . . . . . . . 13 ⊢ (0(,]+∞) ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
80	79, 3	eleqtrri 2839	. . . . . . . . . . . 12 ⊢ (0(,]+∞) ∈ 𝐴
81	72, 80	sselii 3919	. . . . . . . . . . 11 ⊢ (0(,]+∞) ∈ (𝐴 ∪ 𝐵)
82		ssun2 4108	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
83		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (-∞[,)1) = (-∞[,)1)
84		oveq2 7292	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (-∞[,)𝑥) = (-∞[,)1))
85	84	rspceeqv 3576	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ^* ∧ (-∞[,)1) = (-∞[,)1)) → ∃𝑥 ∈ ℝ^* (-∞[,)1) = (-∞[,)𝑥))
86	57, 83, 85	mp2an 689	. . . . . . . . . . . . . 14 ⊢ ∃𝑥 ∈ ℝ^* (-∞[,)1) = (-∞[,)𝑥)
87		ovex 7317	. . . . . . . . . . . . . . 15 ⊢ (-∞[,)𝑥) ∈ V
88	20, 87	elrnmpti 5872	. . . . . . . . . . . . . 14 ⊢ ((-∞[,)1) ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) ↔ ∃𝑥 ∈ ℝ^* (-∞[,)1) = (-∞[,)𝑥))
89	86, 88	mpbir 230	. . . . . . . . . . . . 13 ⊢ (-∞[,)1) ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
90	89, 5	eleqtrri 2839	. . . . . . . . . . . 12 ⊢ (-∞[,)1) ∈ 𝐵
91	82, 90	sselii 3919	. . . . . . . . . . 11 ⊢ (-∞[,)1) ∈ (𝐴 ∪ 𝐵)
92		prssi 4755	. . . . . . . . . . 11 ⊢ (((0(,]+∞) ∈ (𝐴 ∪ 𝐵) ∧ (-∞[,)1) ∈ (𝐴 ∪ 𝐵)) → {(0(,]+∞), (-∞[,)1)} ⊆ (𝐴 ∪ 𝐵))
93	81, 91, 92	mp2an 689	. . . . . . . . . 10 ⊢ {(0(,]+∞), (-∞[,)1)} ⊆ (𝐴 ∪ 𝐵)
94		ssfii 9187	. . . . . . . . . . 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 3931	. . . . . . . . 9 ⊢ {(0(,]+∞), (-∞[,)1)} ⊆ (fi‘(𝐴 ∪ 𝐵))
97		eltg3i 22120	. . . . . . . . 9 ⊢ (((fi‘(𝐴 ∪ 𝐵)) ∈ V ∧ {(0(,]+∞), (-∞[,)1)} ⊆ (fi‘(𝐴 ∪ 𝐵))) → ∪ {(0(,]+∞), (-∞[,)1)} ∈ (topGen‘(fi‘(𝐴 ∪ 𝐵))))
98	71, 96, 97	mp2an 689	. . . . . . . 8 ⊢ ∪ {(0(,]+∞), (-∞[,)1)} ∈ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
99	70, 98	eqeltrri 2837	. . . . . . 7 ⊢ ℝ^* ∈ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
100		snssi 4742	. . . . . . 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 22125	. . . . . . . 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 3931	. . . . . 6 ⊢ (𝐴 ∪ 𝐵) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
105	101, 104	unssi 4120	. . . . 5 ⊢ ({ℝ^*} ∪ (𝐴 ∪ 𝐵)) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
106		fiss 9192	. . . . 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 689	. . . 4 ⊢ (fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))) ⊆ (fi‘(topGen‘(fi‘(𝐴 ∪ 𝐵))))
108		fibas 22136	. . . . 5 ⊢ (fi‘(𝐴 ∪ 𝐵)) ∈ TopBases
109		tgcl 22128	. . . . 5 ⊢ ((fi‘(𝐴 ∪ 𝐵)) ∈ TopBases → (topGen‘(fi‘(𝐴 ∪ 𝐵))) ∈ Top)
110		fitop 22058	. . . . 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 3958	. . 3 ⊢ (fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
113		2basgen 22149	. . 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 689	. 2 ⊢ (topGen‘(fi‘(𝐴 ∪ 𝐵))) = (topGen‘(fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))))
115	8, 114	eqtr4i 2770	1 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘(𝐴 ∪ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∃wrex 3066 Vcvv 3433 ∪ cun 3886 ⊆ wss 3888 𝒫 cpw 4534 {csn 4562 {cpr 4564 ∪ cuni 4840 class class class wbr 5075 ↦ cmpt 5158 ran crn 5591 ⟶wf 6433 ‘cfv 6437 (class class class)co 7284 ficfi 9178 0cc0 10880 1c1 10881 +∞cpnf 11015 -∞cmnf 11016 ℝ^*cxr 11017 < clt 11018 ≤ cle 11019 (,]cioc 13089 [,)cico 13090 [,]cicc 13091 topGenctg 17157 ordTopcordt 17219 TosetRel ctsr 18292 Topctop 22051 TopBasesctb 22104

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2710 ax-sep 5224 ax-nul 5231 ax-pow 5289 ax-pr 5353 ax-un 7597 ax-cnex 10936 ax-resscn 10937 ax-1cn 10938 ax-icn 10939 ax-addcl 10940 ax-addrcl 10941 ax-mulcl 10942 ax-mulrcl 10943 ax-mulcom 10944 ax-addass 10945 ax-mulass 10946 ax-distr 10947 ax-i2m1 10948 ax-1ne0 10949 ax-1rid 10950 ax-rnegex 10951 ax-rrecex 10952 ax-cnre 10953 ax-pre-lttri 10954 ax-pre-lttrn 10955 ax-pre-ltadd 10956 ax-pre-mulgt0 10957

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3073 df-rab 3074 df-v 3435 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-f1 6442 df-fo 6443 df-f1o 6444 df-fv 6445 df-riota 7241 df-ov 7287 df-oprab 7288 df-mpo 7289 df-om 7722 df-1st 7840 df-2nd 7841 df-1o 8306 df-er 8507 df-en 8743 df-dom 8744 df-sdom 8745 df-fin 8746 df-fi 9179 df-pnf 11020 df-mnf 11021 df-xr 11022 df-ltxr 11023 df-le 11024 df-sub 11216 df-neg 11217 df-ioc 13093 df-ico 13094 df-icc 13095 df-topgen 17163 df-ordt 17221 df-ps 18293 df-tsr 18294 df-top 22052 df-bases 22105

This theorem is referenced by: leordtval 22373 lecldbas 22379

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