leordtval2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > leordtval2		Structured version Visualization version GIF version

Theorem leordtval2 21819

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 17836	. . 3 ⊢ ≤ ∈ TosetRel
2		ledm 17833	. . . 4 ⊢ ℝ^* = dom ≤
3		leordtval.1	. . . . 5 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
4	3	leordtvallem1 21817	. . . 4 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
5		leordtval.2	. . . . 5 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
6	3, 5	leordtvallem2 21818	. . . 4 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
7	2, 4, 6	ordtval 21796	. . 3 ⊢ ( ≤ ∈ TosetRel → (ordTop‘ ≤ ) = (topGen‘(fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵)))))
8	1, 7	ax-mp 5	. 2 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))))
9		snex 5331	. . . . 5 ⊢ {ℝ^*} ∈ V
10		xrex 12385	. . . . . . 7 ⊢ ℝ^* ∈ V
11	10	pwex 5280	. . . . . 6 ⊢ 𝒫 ℝ^* ∈ V
12		eqid 2821	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
13		iocssxr 12819	. . . . . . . . . . . 12 ⊢ (𝑥(,]+∞) ⊆ ℝ^*
14	10, 13	elpwi2 5248	. . . . . . . . . . 11 ⊢ (𝑥(,]+∞) ∈ 𝒫 ℝ^*
15	14	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* → (𝑥(,]+∞) ∈ 𝒫 ℝ^*)
16	12, 15	fmpti 6875	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)):ℝ^⟶𝒫 ℝ^
17		frn 6519	. . . . . . . . 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 4000	. . . . . . 7 ⊢ 𝐴 ⊆ 𝒫 ℝ^*
20		eqid 2821	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
21		icossxr 12820	. . . . . . . . . . . 12 ⊢ (-∞[,)𝑥) ⊆ ℝ^*
22	10, 21	elpwi2 5248	. . . . . . . . . . 11 ⊢ (-∞[,)𝑥) ∈ 𝒫 ℝ^*
23	22	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ^* → (-∞[,)𝑥) ∈ 𝒫 ℝ^*)
24	20, 23	fmpti 6875	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)):ℝ^⟶𝒫 ℝ^
25		frn 6519	. . . . . . . . 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 4000	. . . . . . 7 ⊢ 𝐵 ⊆ 𝒫 ℝ^*
28	19, 27	unssi 4160	. . . . . 6 ⊢ (𝐴 ∪ 𝐵) ⊆ 𝒫 ℝ^*
29	11, 28	ssexi 5225	. . . . 5 ⊢ (𝐴 ∪ 𝐵) ∈ V
30	9, 29	unex 7468	. . . 4 ⊢ ({ℝ^*} ∪ (𝐴 ∪ 𝐵)) ∈ V
31		ssun2 4148	. . . 4 ⊢ (𝐴 ∪ 𝐵) ⊆ ({ℝ^*} ∪ (𝐴 ∪ 𝐵))
32		fiss 8887	. . . 4 ⊢ ((({ℝ^} ∪ (𝐴 ∪ 𝐵)) ∈ V ∧ (𝐴 ∪ 𝐵) ⊆ ({ℝ^} ∪ (𝐴 ∪ 𝐵))) → (fi‘(𝐴 ∪ 𝐵)) ⊆ (fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))))
33	30, 31, 32	mp2an 690	. . 3 ⊢ (fi‘(𝐴 ∪ 𝐵)) ⊆ (fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵)))
34		fvex 6682	. . . . 5 ⊢ (topGen‘(fi‘(𝐴 ∪ 𝐵))) ∈ V
35		ovex 7188	. . . . . . . . . 10 ⊢ (0(,]+∞) ∈ V
36		ovex 7188	. . . . . . . . . 10 ⊢ (-∞[,)1) ∈ V
37	35, 36	unipr 4854	. . . . . . . . 9 ⊢ ∪ {(0(,]+∞), (-∞[,)1)} = ((0(,]+∞) ∪ (-∞[,)1))
38		iocssxr 12819	. . . . . . . . . . 11 ⊢ (0(,]+∞) ⊆ ℝ^*
39		icossxr 12820	. . . . . . . . . . 11 ⊢ (-∞[,)1) ⊆ ℝ^*
40	38, 39	unssi 4160	. . . . . . . . . 10 ⊢ ((0(,]+∞) ∪ (-∞[,)1)) ⊆ ℝ^*
41		mnfxr 10697	. . . . . . . . . . . . 13 ⊢ -∞ ∈ ℝ^*
42		0xr 10687	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
43		pnfxr 10694	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
44		mnflt0 12519	. . . . . . . . . . . . . 14 ⊢ -∞ < 0
45		0lepnf 12526	. . . . . . . . . . . . . 14 ⊢ 0 ≤ +∞
46		df-icc 12744	. . . . . . . . . . . . . . 15 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
47		df-ioc 12742	. . . . . . . . . . . . . . 15 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
48		xrltnle 10707	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (0 < 𝑤 ↔ ¬ 𝑤 ≤ 0))
49		xrletr 12550	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 ≤ 0 ∧ 0 ≤ +∞) → 𝑤 ≤ +∞))
50		xrlttr 12532	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < 0 ∧ 0 < 𝑤) → -∞ < 𝑤))
51		xrltle 12541	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (-∞ < 𝑤 → -∞ ≤ 𝑤))
52	51	3adant2 1127	. . . . . . . . . . . . . . . 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 12753	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < 0 ∧ 0 ≤ +∞)) → ((-∞[,]0) ∪ (0(,]+∞)) = (-∞[,]+∞))
55	44, 45, 54	mpanr12 703	. . . . . . . . . . . . 13 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞[,]0) ∪ (0(,]+∞)) = (-∞[,]+∞))
56	41, 42, 43, 55	mp3an 1457	. . . . . . . . . . . 12 ⊢ ((-∞[,]0) ∪ (0(,]+∞)) = (-∞[,]+∞)
57		1xr 10699	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ^*
58		0lt1 11161	. . . . . . . . . . . . . 14 ⊢ 0 < 1
59		df-ico 12743	. . . . . . . . . . . . . . 15 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
60		xrlelttr 12548	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((𝑤 ≤ 0 ∧ 0 < 1) → 𝑤 < 1))
61	59, 46, 60	ixxss2 12756	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ^* ∧ 0 < 1) → (-∞[,]0) ⊆ (-∞[,)1))
62	57, 58, 61	mp2an 690	. . . . . . . . . . . . 13 ⊢ (-∞[,]0) ⊆ (-∞[,)1)
63		unss1 4154	. . . . . . . . . . . . 13 ⊢ ((-∞[,]0) ⊆ (-∞[,)1) → ((-∞[,]0) ∪ (0(,]+∞)) ⊆ ((-∞[,)1) ∪ (0(,]+∞)))
64	62, 63	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((-∞[,]0) ∪ (0(,]+∞)) ⊆ ((-∞[,)1) ∪ (0(,]+∞))
65	56, 64	eqsstrri 4001	. . . . . . . . . . 11 ⊢ (-∞[,]+∞) ⊆ ((-∞[,)1) ∪ (0(,]+∞))
66		iccmax 12811	. . . . . . . . . . 11 ⊢ (-∞[,]+∞) = ℝ^*
67		uncom 4128	. . . . . . . . . . 11 ⊢ ((-∞[,)1) ∪ (0(,]+∞)) = ((0(,]+∞) ∪ (-∞[,)1))
68	65, 66, 67	3sstr3i 4008	. . . . . . . . . 10 ⊢ ℝ^* ⊆ ((0(,]+∞) ∪ (-∞[,)1))
69	40, 68	eqssi 3982	. . . . . . . . 9 ⊢ ((0(,]+∞) ∪ (-∞[,)1)) = ℝ^*
70	37, 69	eqtri 2844	. . . . . . . 8 ⊢ ∪ {(0(,]+∞), (-∞[,)1)} = ℝ^*
71		fvex 6682	. . . . . . . . 9 ⊢ (fi‘(𝐴 ∪ 𝐵)) ∈ V
72		ssun1 4147	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
73		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ (0(,]+∞) = (0(,]+∞)
74		oveq1 7162	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 0 → (𝑥(,]+∞) = (0(,]+∞))
75	74	rspceeqv 3637	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ (0(,]+∞) = (0(,]+∞)) → ∃𝑥 ∈ ℝ^* (0(,]+∞) = (𝑥(,]+∞))
76	42, 73, 75	mp2an 690	. . . . . . . . . . . . . 14 ⊢ ∃𝑥 ∈ ℝ^* (0(,]+∞) = (𝑥(,]+∞)
77		ovex 7188	. . . . . . . . . . . . . . 15 ⊢ (𝑥(,]+∞) ∈ V
78	12, 77	elrnmpti 5831	. . . . . . . . . . . . . 14 ⊢ ((0(,]+∞) ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ↔ ∃𝑥 ∈ ℝ^* (0(,]+∞) = (𝑥(,]+∞))
79	76, 78	mpbir 233	. . . . . . . . . . . . 13 ⊢ (0(,]+∞) ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
80	79, 3	eleqtrri 2912	. . . . . . . . . . . 12 ⊢ (0(,]+∞) ∈ 𝐴
81	72, 80	sselii 3963	. . . . . . . . . . 11 ⊢ (0(,]+∞) ∈ (𝐴 ∪ 𝐵)
82		ssun2 4148	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
83		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ (-∞[,)1) = (-∞[,)1)
84		oveq2 7163	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (-∞[,)𝑥) = (-∞[,)1))
85	84	rspceeqv 3637	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ^* ∧ (-∞[,)1) = (-∞[,)1)) → ∃𝑥 ∈ ℝ^* (-∞[,)1) = (-∞[,)𝑥))
86	57, 83, 85	mp2an 690	. . . . . . . . . . . . . 14 ⊢ ∃𝑥 ∈ ℝ^* (-∞[,)1) = (-∞[,)𝑥)
87		ovex 7188	. . . . . . . . . . . . . . 15 ⊢ (-∞[,)𝑥) ∈ V
88	20, 87	elrnmpti 5831	. . . . . . . . . . . . . 14 ⊢ ((-∞[,)1) ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) ↔ ∃𝑥 ∈ ℝ^* (-∞[,)1) = (-∞[,)𝑥))
89	86, 88	mpbir 233	. . . . . . . . . . . . 13 ⊢ (-∞[,)1) ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
90	89, 5	eleqtrri 2912	. . . . . . . . . . . 12 ⊢ (-∞[,)1) ∈ 𝐵
91	82, 90	sselii 3963	. . . . . . . . . . 11 ⊢ (-∞[,)1) ∈ (𝐴 ∪ 𝐵)
92		prssi 4753	. . . . . . . . . . 11 ⊢ (((0(,]+∞) ∈ (𝐴 ∪ 𝐵) ∧ (-∞[,)1) ∈ (𝐴 ∪ 𝐵)) → {(0(,]+∞), (-∞[,)1)} ⊆ (𝐴 ∪ 𝐵))
93	81, 91, 92	mp2an 690	. . . . . . . . . 10 ⊢ {(0(,]+∞), (-∞[,)1)} ⊆ (𝐴 ∪ 𝐵)
94		ssfii 8882	. . . . . . . . . . 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 3975	. . . . . . . . 9 ⊢ {(0(,]+∞), (-∞[,)1)} ⊆ (fi‘(𝐴 ∪ 𝐵))
97		eltg3i 21568	. . . . . . . . 9 ⊢ (((fi‘(𝐴 ∪ 𝐵)) ∈ V ∧ {(0(,]+∞), (-∞[,)1)} ⊆ (fi‘(𝐴 ∪ 𝐵))) → ∪ {(0(,]+∞), (-∞[,)1)} ∈ (topGen‘(fi‘(𝐴 ∪ 𝐵))))
98	71, 96, 97	mp2an 690	. . . . . . . 8 ⊢ ∪ {(0(,]+∞), (-∞[,)1)} ∈ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
99	70, 98	eqeltrri 2910	. . . . . . 7 ⊢ ℝ^* ∈ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
100		snssi 4740	. . . . . . 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 21573	. . . . . . . 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 3975	. . . . . 6 ⊢ (𝐴 ∪ 𝐵) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
105	101, 104	unssi 4160	. . . . 5 ⊢ ({ℝ^*} ∪ (𝐴 ∪ 𝐵)) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
106		fiss 8887	. . . . 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 690	. . . 4 ⊢ (fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))) ⊆ (fi‘(topGen‘(fi‘(𝐴 ∪ 𝐵))))
108		fibas 21584	. . . . 5 ⊢ (fi‘(𝐴 ∪ 𝐵)) ∈ TopBases
109		tgcl 21576	. . . . 5 ⊢ ((fi‘(𝐴 ∪ 𝐵)) ∈ TopBases → (topGen‘(fi‘(𝐴 ∪ 𝐵))) ∈ Top)
110		fitop 21507	. . . . 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 4002	. . 3 ⊢ (fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))) ⊆ (topGen‘(fi‘(𝐴 ∪ 𝐵)))
113		2basgen 21597	. . 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 690	. 2 ⊢ (topGen‘(fi‘(𝐴 ∪ 𝐵))) = (topGen‘(fi‘({ℝ^*} ∪ (𝐴 ∪ 𝐵))))
115	8, 114	eqtr4i 2847	1 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘(𝐴 ∪ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 Vcvv 3494 ∪ cun 3933 ⊆ wss 3935 𝒫 cpw 4538 {csn 4566 {cpr 4568 ∪ cuni 4837 class class class wbr 5065 ↦ cmpt 5145 ran crn 5555 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ficfi 8873 0cc0 10536 1c1 10537 +∞cpnf 10671 -∞cmnf 10672 ℝ^*cxr 10673 < clt 10674 ≤ cle 10675 (,]cioc 12738 [,)cico 12739 [,]cicc 12740 topGenctg 16710 ordTopcordt 16771 TosetRel ctsr 17808 Topctop 21500 TopBasesctb 21552

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fi 8874 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-ioc 12742 df-ico 12743 df-icc 12744 df-topgen 16716 df-ordt 16773 df-ps 17809 df-tsr 17810 df-top 21501 df-bases 21553

This theorem is referenced by: leordtval 21820 lecldbas 21826

W3C validator