ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xrltso GIF version

Theorem xrltso 9550
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.
StepHypRef Expression
1 xrltnr 9534 . . . . 5 (𝑥 ∈ ℝ* → ¬ 𝑥 < 𝑥)
21adantl 275 . . . 4 ((⊤ ∧ 𝑥 ∈ ℝ*) → ¬ 𝑥 < 𝑥)
3 xrlttr 9549 . . . . 5 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
43adantl 275 . . . 4 ((⊤ ∧ (𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*)) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
52, 4ispod 4196 . . 3 (⊤ → < Po ℝ*)
65mptru 1325 . 2 < Po ℝ*
7 elxr 9531 . . . . 5 (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞))
8 elxr 9531 . . . . . . . . . 10 (𝑦 ∈ ℝ* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
9 elxr 9531 . . . . . . . . . . . . . 14 (𝑧 ∈ ℝ* ↔ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞))
10 simplr 504 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑥 ∈ ℝ)
11 simpll 503 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ)
12 simpr 109 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
13 axltwlin 7800 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
1410, 11, 12, 13syl3anc 1201 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
15 ltpnf 9535 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℝ → 𝑥 < +∞)
1615ad2antlr 480 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < +∞)
17 breq2 3903 . . . . . . . . . . . . . . . . . . 19 (𝑧 = +∞ → (𝑥 < 𝑧𝑥 < +∞))
1817adantl 275 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑥 < +∞))
1916, 18mpbird 166 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < 𝑧)
2019orcd 707 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑧 < 𝑦))
2120a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
22 mnflt 9537 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ → -∞ < 𝑦)
2322ad2antrr 479 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → -∞ < 𝑦)
24 breq1 3902 . . . . . . . . . . . . . . . . . . 19 (𝑧 = -∞ → (𝑧 < 𝑦 ↔ -∞ < 𝑦))
2524adantl 275 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑧 < 𝑦 ↔ -∞ < 𝑦))
2623, 25mpbird 166 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
2726olcd 708 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑥 < 𝑧𝑧 < 𝑦))
2827a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
2914, 21, 283jaodan 1269 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
309, 29sylan2b 285 . . . . . . . . . . . . 13 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
3130anasss 396 . . . . . . . . . . . 12 ((𝑦 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
3231ancoms 266 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
33 ltpnf 9535 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℝ → 𝑧 < +∞)
3433adantl 275 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 < +∞)
35 breq2 3903 . . . . . . . . . . . . . . . . . . 19 (𝑦 = +∞ → (𝑧 < 𝑦𝑧 < +∞))
3635ad2antrr 479 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑧 < 𝑦𝑧 < +∞))
3734, 36mpbird 166 . . . . . . . . . . . . . . . . 17 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 < 𝑦)
3837olcd 708 . . . . . . . . . . . . . . . 16 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑧𝑧 < 𝑦))
3938a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
4015ad2antlr 480 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < +∞)
4117adantl 275 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑥 < +∞))
4240, 41mpbird 166 . . . . . . . . . . . . . . . . 17 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < 𝑧)
4342orcd 707 . . . . . . . . . . . . . . . 16 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑧 < 𝑦))
4443a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
45 mnfltpnf 9539 . . . . . . . . . . . . . . . . . . 19 -∞ < +∞
46 breq12 3904 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 = -∞ ∧ 𝑦 = +∞) → (𝑧 < 𝑦 ↔ -∞ < +∞))
4746ancoms 266 . . . . . . . . . . . . . . . . . . 19 ((𝑦 = +∞ ∧ 𝑧 = -∞) → (𝑧 < 𝑦 ↔ -∞ < +∞))
4845, 47mpbiri 167 . . . . . . . . . . . . . . . . . 18 ((𝑦 = +∞ ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
4948adantlr 468 . . . . . . . . . . . . . . . . 17 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
5049olcd 708 . . . . . . . . . . . . . . . 16 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑥 < 𝑧𝑧 < 𝑦))
5150a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
5239, 44, 513jaodan 1269 . . . . . . . . . . . . . 14 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
539, 52sylan2b 285 . . . . . . . . . . . . 13 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
5453anasss 396 . . . . . . . . . . . 12 ((𝑦 = +∞ ∧ (𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
5554ancoms 266 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
56 rexr 7779 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
57 nltmnf 9542 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
5856, 57syl 14 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → ¬ 𝑥 < -∞)
5958ad2antrr 479 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → ¬ 𝑥 < -∞)
60 breq2 3903 . . . . . . . . . . . . . 14 (𝑦 = -∞ → (𝑥 < 𝑦𝑥 < -∞))
6160adantl 275 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → (𝑥 < 𝑦𝑥 < -∞))
6259, 61mtbird 647 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → ¬ 𝑥 < 𝑦)
6362pm2.21d 593 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
6432, 55, 633jaodan 1269 . . . . . . . . . 10 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
658, 64sylan2b 285 . . . . . . . . 9 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
6665anasss 396 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ (𝑧 ∈ ℝ*𝑦 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
6766ancoms 266 . . . . . . 7 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
68 pnfnlt 9541 . . . . . . . . . 10 (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
6968ad2antlr 480 . . . . . . . . 9 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → ¬ +∞ < 𝑦)
70 breq1 3902 . . . . . . . . . 10 (𝑥 = +∞ → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
7170adantl 275 . . . . . . . . 9 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
7269, 71mtbird 647 . . . . . . . 8 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → ¬ 𝑥 < 𝑦)
7372pm2.21d 593 . . . . . . 7 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
74 df-3or 948 . . . . . . . . . . 11 ((𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞) ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞))
759, 74bitri 183 . . . . . . . . . 10 (𝑧 ∈ ℝ* ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞))
76 mnfltxr 9540 . . . . . . . . . . . . . . 15 ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) → -∞ < 𝑧)
7776adantl 275 . . . . . . . . . . . . . 14 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → -∞ < 𝑧)
78 breq1 3902 . . . . . . . . . . . . . . 15 (𝑥 = -∞ → (𝑥 < 𝑧 ↔ -∞ < 𝑧))
7978adantr 274 . . . . . . . . . . . . . 14 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑧 ↔ -∞ < 𝑧))
8077, 79mpbird 166 . . . . . . . . . . . . 13 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → 𝑥 < 𝑧)
8180orcd 707 . . . . . . . . . . . 12 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑧𝑧 < 𝑦))
8281a1d 22 . . . . . . . . . . 11 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
83 eqtr3 2137 . . . . . . . . . . . . 13 ((𝑥 = -∞ ∧ 𝑧 = -∞) → 𝑥 = 𝑧)
8483breq1d 3909 . . . . . . . . . . . 12 ((𝑥 = -∞ ∧ 𝑧 = -∞) → (𝑥 < 𝑦𝑧 < 𝑦))
85 olc 685 . . . . . . . . . . . 12 (𝑧 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦))
8684, 85syl6bi 162 . . . . . . . . . . 11 ((𝑥 = -∞ ∧ 𝑧 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
8782, 86jaodan 771 . . . . . . . . . 10 ((𝑥 = -∞ ∧ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
8875, 87sylan2b 285 . . . . . . . . 9 ((𝑥 = -∞ ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
8988ancoms 266 . . . . . . . 8 ((𝑧 ∈ ℝ*𝑥 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
9089adantlr 468 . . . . . . 7 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
9167, 73, 903jaodan 1269 . . . . . 6 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
92913impa 1161 . . . . 5 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
937, 92syl3an3b 1239 . . . 4 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ*𝑥 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
94933com13 1171 . . 3 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
9594rgen3 2496 . 2 𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ* (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦))
96 df-iso 4189 . 2 ( < Or ℝ* ↔ ( < Po ℝ* ∧ ∀𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ* (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦))))
976, 95, 96mpbir2an 911 1 < Or ℝ*
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 682  w3o 946  w3a 947   = wceq 1316  wtru 1317  wcel 1465  wral 2393   class class class wbr 3899   Po wpo 4186   Or wor 4187  cr 7587  +∞cpnf 7765  -∞cmnf 7766  *cxr 7767   < clt 7768
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-po 4188  df-iso 4189  df-xp 4515  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773
This theorem is referenced by:  xrlelttr  9557  xrltletr  9558  xrletr  9559  xrmaxiflemlub  10985
  Copyright terms: Public domain W3C validator