Theorem xrltso 9783

Description: 'Less than' is a weakly linear ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)

Assertion

Ref	Expression
xrltso	⊢ < Or ℝ*

Proof of Theorem xrltso

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrltnr 9766	. . . . 5 ⊢ (𝑥 ∈ ℝ* → ¬ 𝑥 < 𝑥)
2	1	adantl 277	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ*) → ¬ 𝑥 < 𝑥)
3		xrlttr 9782	. . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧))
4	3	adantl 277	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*)) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧))
5	2, 4	ispod 4301	. . 3 ⊢ (⊤ → < Po ℝ*)
6	5	mptru 1362	. 2 ⊢ < Po ℝ*
7		elxr 9763	. . . . 5 ⊢ (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞))
8		elxr 9763	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
9		elxr 9763	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℝ* ↔ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞))
10		simplr 528	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑥 ∈ ℝ)
11		simpll 527	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ)
12		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
13		axltwlin 8015	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
14	10, 11, 12, 13	syl3anc 1238	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
15		ltpnf 9767	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞)
16	15	ad2antlr 489	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < +∞)
17		breq2 4004	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = +∞ → (𝑥 < 𝑧 ↔ 𝑥 < +∞))
18	17	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧 ↔ 𝑥 < +∞))
19	16, 18	mpbird 167	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < 𝑧)
20	19	orcd 733	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))
21	20	a1d 22	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
22		mnflt 9770	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ → -∞ < 𝑦)
23	22	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → -∞ < 𝑦)
24		breq1 4003	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = -∞ → (𝑧 < 𝑦 ↔ -∞ < 𝑦))
25	24	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑧 < 𝑦 ↔ -∞ < 𝑦))
26	23, 25	mpbird 167	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
27	26	olcd 734	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))
28	27	a1d 22	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
29	14, 21, 28	3jaodan 1306	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
30	9, 29	sylan2b 287	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
31	30	anasss 399	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
32	31	ancoms 268	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
33		ltpnf 9767	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℝ → 𝑧 < +∞)
34	33	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 < +∞)
35		breq2 4004	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = +∞ → (𝑧 < 𝑦 ↔ 𝑧 < +∞))
36	35	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑧 < 𝑦 ↔ 𝑧 < +∞))
37	34, 36	mpbird 167	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 < 𝑦)
38	37	olcd 734	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))
39	38	a1d 22	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
40	15	ad2antlr 489	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < +∞)
41	17	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧 ↔ 𝑥 < +∞))
42	40, 41	mpbird 167	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < 𝑧)
43	42	orcd 733	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))
44	43	a1d 22	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
45		mnfltpnf 9772	. . . . . . . . . . . . . . . . . . 19 ⊢ -∞ < +∞
46		breq12 4005	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 = -∞ ∧ 𝑦 = +∞) → (𝑧 < 𝑦 ↔ -∞ < +∞))
47	46	ancoms 268	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 = +∞ ∧ 𝑧 = -∞) → (𝑧 < 𝑦 ↔ -∞ < +∞))
48	45, 47	mpbiri 168	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 = +∞ ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
49	48	adantlr 477	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
50	49	olcd 734	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))
51	50	a1d 22	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
52	39, 44, 51	3jaodan 1306	. . . . . . . . . . . . . 14 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
53	9, 52	sylan2b 287	. . . . . . . . . . . . 13 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
54	53	anasss 399	. . . . . . . . . . . 12 ⊢ ((𝑦 = +∞ ∧ (𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
55	54	ancoms 268	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
56		rexr 7993	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
57		nltmnf 9775	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
58	56, 57	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 < -∞)
59	58	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → ¬ 𝑥 < -∞)
60		breq2 4004	. . . . . . . . . . . . . 14 ⊢ (𝑦 = -∞ → (𝑥 < 𝑦 ↔ 𝑥 < -∞))
61	60	adantl 277	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → (𝑥 < 𝑦 ↔ 𝑥 < -∞))
62	59, 61	mtbird 673	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → ¬ 𝑥 < 𝑦)
63	62	pm2.21d 619	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
64	32, 55, 63	3jaodan 1306	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
65	8, 64	sylan2b 287	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
66	65	anasss 399	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ (𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
67	66	ancoms 268	. . . . . . 7 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
68		pnfnlt 9774	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
69	68	ad2antlr 489	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → ¬ +∞ < 𝑦)
70		breq1 4003	. . . . . . . . . 10 ⊢ (𝑥 = +∞ → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
71	70	adantl 277	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
72	69, 71	mtbird 673	. . . . . . . 8 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → ¬ 𝑥 < 𝑦)
73	72	pm2.21d 619	. . . . . . 7 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
74		df-3or 979	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞) ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞))
75	9, 74	bitri 184	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ* ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞))
76		mnfltxr 9773	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) → -∞ < 𝑧)
77	76	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → -∞ < 𝑧)
78		breq1 4003	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = -∞ → (𝑥 < 𝑧 ↔ -∞ < 𝑧))
79	78	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑧 ↔ -∞ < 𝑧))
80	77, 79	mpbird 167	. . . . . . . . . . . . 13 ⊢ ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → 𝑥 < 𝑧)
81	80	orcd 733	. . . . . . . . . . . 12 ⊢ ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))
82	81	a1d 22	. . . . . . . . . . 11 ⊢ ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
83		eqtr3 2197	. . . . . . . . . . . . 13 ⊢ ((𝑥 = -∞ ∧ 𝑧 = -∞) → 𝑥 = 𝑧)
84	83	breq1d 4010	. . . . . . . . . . . 12 ⊢ ((𝑥 = -∞ ∧ 𝑧 = -∞) → (𝑥 < 𝑦 ↔ 𝑧 < 𝑦))
85		olc 711	. . . . . . . . . . . 12 ⊢ (𝑧 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))
86	84, 85	syl6bi 163	. . . . . . . . . . 11 ⊢ ((𝑥 = -∞ ∧ 𝑧 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
87	82, 86	jaodan 797	. . . . . . . . . 10 ⊢ ((𝑥 = -∞ ∧ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
88	75, 87	sylan2b 287	. . . . . . . . 9 ⊢ ((𝑥 = -∞ ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
89	88	ancoms 268	. . . . . . . 8 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑥 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
90	89	adantlr 477	. . . . . . 7 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
91	67, 73, 90	3jaodan 1306	. . . . . 6 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
92	91	3impa 1194	. . . . 5 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
93	7, 92	syl3an3b 1276	. . . 4 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑥 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
94	93	3com13 1208	. . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
95	94	rgen3 2564	. 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* ∀𝑧 ∈ ℝ* (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))
96		df-iso 4294	. 2 ⊢ ( < Or ℝ* ↔ ( < Po ℝ* ∧ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* ∀𝑧 ∈ ℝ* (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))))
97	6, 95, 96	mpbir2an 942	1 ⊢ < Or ℝ*

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   ∨ w3o 977   ∧ w3a 978   = wceq 1353  ⊤wtru 1354   ∈ wcel 2148  ∀wral 2455   class class class wbr 4000   Po wpo 4291   Or wor 4292  ℝcr 7801  +∞cpnf 7979  -∞cmnf 7980  ℝ^*cxr 7981   < clt 7982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-po 4293  df-iso 4294  df-xp 4629  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987

This theorem is referenced by:  xrlelttr  9793  xrltletr  9794  xrletr  9795  xrmaxiflemlub  11240