Theorem xrltso 9582

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 9566	. . . . 5 ⊢ (𝑥 ∈ ℝ* → ¬ 𝑥 < 𝑥)
2	1	adantl 275	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ*) → ¬ 𝑥 < 𝑥)
3		xrlttr 9581	. . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧))
4	3	adantl 275	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*)) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧))
5	2, 4	ispod 4226	. . 3 ⊢ (⊤ → < Po ℝ*)
6	5	mptru 1340	. 2 ⊢ < Po ℝ*
7		elxr 9563	. . . . 5 ⊢ (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞))
8		elxr 9563	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
9		elxr 9563	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℝ* ↔ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞))
10		simplr 519	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑥 ∈ ℝ)
11		simpll 518	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ)
12		simpr 109	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
13		axltwlin 7832	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
14	10, 11, 12, 13	syl3anc 1216	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
15		ltpnf 9567	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞)
16	15	ad2antlr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < +∞)
17		breq2 3933	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = +∞ → (𝑥 < 𝑧 ↔ 𝑥 < +∞))
18	17	adantl 275	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧 ↔ 𝑥 < +∞))
19	16, 18	mpbird 166	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < 𝑧)
20	19	orcd 722	. . . . . . . . . . . . . . . 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 9569	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ → -∞ < 𝑦)
23	22	ad2antrr 479	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → -∞ < 𝑦)
24		breq1 3932	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = -∞ → (𝑧 < 𝑦 ↔ -∞ < 𝑦))
25	24	adantl 275	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑧 < 𝑦 ↔ -∞ < 𝑦))
26	23, 25	mpbird 166	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
27	26	olcd 723	. . . . . . . . . . . . . . . 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 1284	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
30	9, 29	sylan2b 285	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
31	30	anasss 396	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
32	31	ancoms 266	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
33		ltpnf 9567	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℝ → 𝑧 < +∞)
34	33	adantl 275	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 < +∞)
35		breq2 3933	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = +∞ → (𝑧 < 𝑦 ↔ 𝑧 < +∞))
36	35	ad2antrr 479	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑧 < 𝑦 ↔ 𝑧 < +∞))
37	34, 36	mpbird 166	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 < 𝑦)
38	37	olcd 723	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < +∞)
41	17	adantl 275	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧 ↔ 𝑥 < +∞))
42	40, 41	mpbird 166	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < 𝑧)
43	42	orcd 722	. . . . . . . . . . . . . . . 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 9571	. . . . . . . . . . . . . . . . . . 19 ⊢ -∞ < +∞
46		breq12 3934	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 = -∞ ∧ 𝑦 = +∞) → (𝑧 < 𝑦 ↔ -∞ < +∞))
47	46	ancoms 266	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 = +∞ ∧ 𝑧 = -∞) → (𝑧 < 𝑦 ↔ -∞ < +∞))
48	45, 47	mpbiri 167	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 = +∞ ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
49	48	adantlr 468	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
50	49	olcd 723	. . . . . . . . . . . . . . . 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 1284	. . . . . . . . . . . . . 14 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
53	9, 52	sylan2b 285	. . . . . . . . . . . . 13 ⊢ (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
54	53	anasss 396	. . . . . . . . . . . 12 ⊢ ((𝑦 = +∞ ∧ (𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
55	54	ancoms 266	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
56		rexr 7811	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
57		nltmnf 9574	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
58	56, 57	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 < -∞)
59	58	ad2antrr 479	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → ¬ 𝑥 < -∞)
60		breq2 3933	. . . . . . . . . . . . . 14 ⊢ (𝑦 = -∞ → (𝑥 < 𝑦 ↔ 𝑥 < -∞))
61	60	adantl 275	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → (𝑥 < 𝑦 ↔ 𝑥 < -∞))
62	59, 61	mtbird 662	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → ¬ 𝑥 < 𝑦)
63	62	pm2.21d 608	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
64	32, 55, 63	3jaodan 1284	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
65	8, 64	sylan2b 285	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
66	65	anasss 396	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ (𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
67	66	ancoms 266	. . . . . . 7 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
68		pnfnlt 9573	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
69	68	ad2antlr 480	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → ¬ +∞ < 𝑦)
70		breq1 3932	. . . . . . . . . 10 ⊢ (𝑥 = +∞ → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
71	70	adantl 275	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
72	69, 71	mtbird 662	. . . . . . . 8 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → ¬ 𝑥 < 𝑦)
73	72	pm2.21d 608	. . . . . . 7 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
74		df-3or 963	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞) ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞))
75	9, 74	bitri 183	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ* ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞))
76		mnfltxr 9572	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) → -∞ < 𝑧)
77	76	adantl 275	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → -∞ < 𝑧)
78		breq1 3932	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = -∞ → (𝑥 < 𝑧 ↔ -∞ < 𝑧))
79	78	adantr 274	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑧 ↔ -∞ < 𝑧))
80	77, 79	mpbird 166	. . . . . . . . . . . . 13 ⊢ ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → 𝑥 < 𝑧)
81	80	orcd 722	. . . . . . . . . . . 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 2159	. . . . . . . . . . . . 13 ⊢ ((𝑥 = -∞ ∧ 𝑧 = -∞) → 𝑥 = 𝑧)
84	83	breq1d 3939	. . . . . . . . . . . 12 ⊢ ((𝑥 = -∞ ∧ 𝑧 = -∞) → (𝑥 < 𝑦 ↔ 𝑧 < 𝑦))
85		olc 700	. . . . . . . . . . . 12 ⊢ (𝑧 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))
86	84, 85	syl6bi 162	. . . . . . . . . . 11 ⊢ ((𝑥 = -∞ ∧ 𝑧 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
87	82, 86	jaodan 786	. . . . . . . . . 10 ⊢ ((𝑥 = -∞ ∧ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
88	75, 87	sylan2b 285	. . . . . . . . 9 ⊢ ((𝑥 = -∞ ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
89	88	ancoms 266	. . . . . . . 8 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑥 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
90	89	adantlr 468	. . . . . . 7 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
91	67, 73, 90	3jaodan 1284	. . . . . 6 ⊢ (((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
92	91	3impa 1176	. . . . 5 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
93	7, 92	syl3an3b 1254	. . . 4 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑥 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
94	93	3com13 1186	. . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
95	94	rgen3 2519	. 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* ∀𝑧 ∈ ℝ* (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))
96		df-iso 4219	. 2 ⊢ ( < Or ℝ* ↔ ( < Po ℝ* ∧ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* ∀𝑧 ∈ ℝ* (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))))
97	6, 95, 96	mpbir2an 926	1 ⊢ < Or ℝ*

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   ∨ w3o 961   ∧ w3a 962   = wceq 1331  ⊤wtru 1332   ∈ wcel 1480  ∀wral 2416   class class class wbr 3929   Po wpo 4216   Or wor 4217  ℝcr 7619  +∞cpnf 7797  -∞cmnf 7798  ℝ^*cxr 7799   < clt 7800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-po 4218  df-iso 4219  df-xp 4545  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805

This theorem is referenced by:  xrlelttr  9589  xrltletr  9590  xrletr  9591  xrmaxiflemlub  11017