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Theorem leordtval2 23220

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 18638	. . 3 ⊢ ≤ ∈ TosetRel
2		ledm 18635	. . . 4 ⊢ ℝ^* = dom ≤
3		leordtval.1	. . . . 5 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
4	3	leordtvallem1 23218	. . . 4 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
5		leordtval.2	. . . . 5 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
6	3, 5	leordtvallem2 23219	. . . 4 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
7	2, 4, 6	ordtval 23197	. . 3 ⊢ ( ≤ ∈ TosetRel → (ordTop‘ ≤ ) = (topGen‘(fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵)))))
8	1, 7	ax-mp 5	. 2 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))))
9		snex 5436	. . . . 5 ⊢ {ℝ^*} ∈ V
10		xrex 13029	. . . . . . 7 ⊢ ℝ^* ∈ V
11	10	pwex 5380	. . . . . 6 ⊢ 𝒫 ℝ^* ∈ V
12		eqid 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
13		iocssxr 13471	. . . . . . . . . . . 12 ⊢ (𝑥(,]+∞) ⊆ ℝ^*
14	10, 13	elpwi2 5335	. . . . . . . . . . 11 ⊢ (𝑥(,]+∞) ∈ 𝒫 ℝ^*
15	14	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* → (𝑥(,]+∞) ∈ 𝒫 ℝ^*)
16	12, 15	fmpti 7132	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)):ℝ^⟶𝒫 ℝ^
17		frn 6743	. . . . . . . . 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 4030	. . . . . . 7 ⊢ 𝐴 ⊆ 𝒫 ℝ^*
20		eqid 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
21		icossxr 13472	. . . . . . . . . . . 12 ⊢ (-∞[,)𝑥) ⊆ ℝ^*
22	10, 21	elpwi2 5335	. . . . . . . . . . 11 ⊢ (-∞[,)𝑥) ∈ 𝒫 ℝ^*
23	22	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* → (-∞[,)𝑥) ∈ 𝒫 ℝ^*)
24	20, 23	fmpti 7132	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)):ℝ^⟶𝒫 ℝ^
25		frn 6743	. . . . . . . . 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 4030	. . . . . . 7 ⊢ 𝐵 ⊆ 𝒫 ℝ^*
28	19, 27	unssi 4191	. . . . . 6 ⊢ (𝐴 ∪ 𝐵) ⊆ 𝒫 ℝ^*
29	11, 28	ssexi 5322	. . . . 5 ⊢ (𝐴 ∪ 𝐵) ∈ V
30	9, 29	unex 7764	. . . 4 ⊢ ({ℝ^*} ∪ (𝐴 ∪ 𝐵)) ∈ V
31		ssun2 4179	. . . 4 ⊢ (𝐴 ∪ 𝐵) ⊆ ({ℝ^*} ∪ (𝐴 ∪ 𝐵))
32		fiss 9464	. . . 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 6919	. . . . 5 ⊢ (topGen‘(fi‘(𝐴 ∪ 𝐵))) ∈ V
35		ovex 7464	. . . . . . . . . 10 ⊢ (0(,]+∞) ∈ V
36		ovex 7464	. . . . . . . . . 10 ⊢ (-∞[,)1) ∈ V
37	35, 36	unipr 4924	. . . . . . . . 9 ⊢ ∪ {(0(,]+∞), (-∞[,)1)} = ((0(,]+∞) ∪ (-∞[,)1))
38		iocssxr 13471	. . . . . . . . . . 11 ⊢ (0(,]+∞) ⊆ ℝ^*
39		icossxr 13472	. . . . . . . . . . 11 ⊢ (-∞[,)1) ⊆ ℝ^*
40	38, 39	unssi 4191	. . . . . . . . . 10 ⊢ ((0(,]+∞) ∪ (-∞[,)1)) ⊆ ℝ^*
41		mnfxr 11318	. . . . . . . . . . . . 13 ⊢ -∞ ∈ ℝ^*
42		0xr 11308	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
43		pnfxr 11315	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
44		mnflt0 13167	. . . . . . . . . . . . . 14 ⊢ -∞ < 0
45		0lepnf 13175	. . . . . . . . . . . . . 14 ⊢ 0 ≤ +∞
46		df-icc 13394	. . . . . . . . . . . . . . 15 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
47		df-ioc 13392	. . . . . . . . . . . . . . 15 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
48		xrltnle 11328	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (0 < 𝑤 ↔ ¬ 𝑤 ≤ 0))
49		xrletr 13200	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 ≤ 0 ∧ 0 ≤ +∞) → 𝑤 ≤ +∞))
50		xrlttr 13182	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < 0 ∧ 0 < 𝑤) → -∞ < 𝑤))
51		xrltle 13191	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (-∞ < 𝑤 → -∞ ≤ 𝑤))
52	51	3adant2 1132	. . . . . . . . . . . . . . . 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 13403	. . . . . . . . . . . . . 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 11320	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ^*
58		0lt1 11785	. . . . . . . . . . . . . 14 ⊢ 0 < 1
59		df-ico 13393	. . . . . . . . . . . . . . 15 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
60		xrlelttr 13198	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((𝑤 ≤ 0 ∧ 0 < 1) → 𝑤 < 1))
61	59, 46, 60	ixxss2 13406	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ^* ∧ 0 < 1) → (-∞[,]0) ⊆ (-∞[,)1))
62	57, 58, 61	mp2an 692	. . . . . . . . . . . . 13 ⊢ (-∞[,]0) ⊆ (-∞[,)1)
63		unss1 4185	. . . . . . . . . . . . 13 ⊢ ((-∞[,]0) ⊆ (-∞[,)1) → ((-∞[,]0) ∪ (0(,]+∞)) ⊆ ((-∞[,)1) ∪ (0(,]+∞)))
64	62, 63	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((-∞[,]0) ∪ (0(,]+∞)) ⊆ ((-∞[,)1) ∪ (0(,]+∞))
65	56, 64	eqsstrri 4031	. . . . . . . . . . 11 ⊢ (-∞[,]+∞) ⊆ ((-∞[,)1) ∪ (0(,]+∞))
66		iccmax 13463	. . . . . . . . . . 11 ⊢ (-∞[,]+∞) = ℝ^*
67		uncom 4158	. . . . . . . . . . 11 ⊢ ((-∞[,)1) ∪ (0(,]+∞)) = ((0(,]+∞) ∪ (-∞[,)1))
68	65, 66, 67	3sstr3i 4034	. . . . . . . . . 10 ⊢ ℝ^* ⊆ ((0(,]+∞) ∪ (-∞[,)1))
69	40, 68	eqssi 4000	. . . . . . . . 9 ⊢ ((0(,]+∞) ∪ (-∞[,)1)) = ℝ^*
70	37, 69	eqtri 2765	. . . . . . . 8 ⊢ ∪ {(0(,]+∞), (-∞[,)1)} = ℝ^*
71		fvex 6919	. . . . . . . . 9 ⊢ (fi‘(𝐴 ∪ 𝐵)) ∈ V
72		ssun1 4178	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
73		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (0(,]+∞) = (0(,]+∞)
74		oveq1 7438	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 0 → (𝑥(,]+∞) = (0(,]+∞))
75	74	rspceeqv 3645	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ (0(,]+∞) = (0(,]+∞)) → ∃𝑥 ∈ ℝ^* (0(,]+∞) = (𝑥(,]+∞))
76	42, 73, 75	mp2an 692	. . . . . . . . . . . . . 14 ⊢ ∃𝑥 ∈ ℝ^* (0(,]+∞) = (𝑥(,]+∞)
77		ovex 7464	. . . . . . . . . . . . . . 15 ⊢ (𝑥(,]+∞) ∈ V
78	12, 77	elrnmpti 5973	. . . . . . . . . . . . . 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 2840	. . . . . . . . . . . 12 ⊢ (0(,]+∞) ∈ 𝐴
81	72, 80	sselii 3980	. . . . . . . . . . 11 ⊢ (0(,]+∞) ∈ (𝐴 ∪ 𝐵)
82		ssun2 4179	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
83		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (-∞[,)1) = (-∞[,)1)
84		oveq2 7439	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (-∞[,)𝑥) = (-∞[,)1))
85	84	rspceeqv 3645	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ^* ∧ (-∞[,)1) = (-∞[,)1)) → ∃𝑥 ∈ ℝ^* (-∞[,)1) = (-∞[,)𝑥))
86	57, 83, 85	mp2an 692	. . . . . . . . . . . . . 14 ⊢ ∃𝑥 ∈ ℝ^* (-∞[,)1) = (-∞[,)𝑥)
87		ovex 7464	. . . . . . . . . . . . . . 15 ⊢ (-∞[,)𝑥) ∈ V
88	20, 87	elrnmpti 5973	. . . . . . . . . . . . . 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 2840	. . . . . . . . . . . 12 ⊢ (-∞[,)1) ∈ 𝐵
91	82, 90	sselii 3980	. . . . . . . . . . 11 ⊢ (-∞[,)1) ∈ (𝐴 ∪ 𝐵)
92		prssi 4821	. . . . . . . . . . 11 ⊢ (((0(,]+∞) ∈ (𝐴 ∪ 𝐵) ∧ (-∞[,)1) ∈ (𝐴 ∪ 𝐵)) → {(0(,]+∞), (-∞[,)1)} ⊆ (𝐴 ∪ 𝐵))
93	81, 91, 92	mp2an 692	. . . . . . . . . 10 ⊢ {(0(,]+∞), (-∞[,)1)} ⊆ (𝐴 ∪ 𝐵)
94		ssfii 9459	. . . . . . . . . . 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 3993	. . . . . . . . 9 ⊢ {(0(,]+∞), (-∞[,)1)} ⊆ (fi‘(𝐴 ∪ 𝐵))
97		eltg3i 22968	. . . . . . . . 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 2838	. . . . . . 7 ⊢ ℝ^* ∈ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
100		snssi 4808	. . . . . . 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 22973	. . . . . . . 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 3993	. . . . . 6 ⊢ (𝐴 ∪ 𝐵) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
105	101, 104	unssi 4191	. . . . 5 ⊢ ({ℝ^*} ∪ (𝐴 ∪ 𝐵)) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
106		fiss 9464	. . . . 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 22984	. . . . 5 ⊢ (fi‘(𝐴 ∪ 𝐵)) ∈ TopBases
109		tgcl 22976	. . . . 5 ⊢ ((fi‘(𝐴 ∪ 𝐵)) ∈ TopBases → (topGen‘(fi‘(𝐴 ∪ 𝐵))) ∈ Top)
110		fitop 22906	. . . . 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 4032	. . 3 ⊢ (fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
113		2basgen 22997	. . 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 2768	1 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘(𝐴 ∪ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 𝒫 cpw 4600 {csn 4626 {cpr 4628 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ficfi 9450 0cc0 11155 1c1 11156 +∞cpnf 11292 -∞cmnf 11293 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 (,]cioc 13388 [,)cico 13389 [,]cicc 13390 topGenctg 17482 ordTopcordt 17544 TosetRel ctsr 18610 Topctop 22899 TopBasesctb 22952

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-ioc 13392 df-ico 13393 df-icc 13394 df-topgen 17488 df-ordt 17546 df-ps 18611 df-tsr 18612 df-top 22900 df-bases 22953

This theorem is referenced by: leordtval 23221 lecldbas 23227

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