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Theorem xrltso 9198
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 9182 . . . . 5 (𝑥 ∈ ℝ* → ¬ 𝑥 < 𝑥)
21adantl 271 . . . 4 ((⊤ ∧ 𝑥 ∈ ℝ*) → ¬ 𝑥 < 𝑥)
3 xrlttr 9197 . . . . 5 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
43adantl 271 . . . 4 ((⊤ ∧ (𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*)) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
52, 4ispod 4105 . . 3 (⊤ → < Po ℝ*)
65trud 1296 . 2 < Po ℝ*
7 elxr 9179 . . . . 5 (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞))
8 elxr 9179 . . . . . . . . . 10 (𝑦 ∈ ℝ* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
9 elxr 9179 . . . . . . . . . . . . . 14 (𝑧 ∈ ℝ* ↔ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞))
10 simplr 497 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑥 ∈ ℝ)
11 simpll 496 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ)
12 simpr 108 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
13 axltwlin 7498 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
1410, 11, 12, 13syl3anc 1172 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
15 ltpnf 9183 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℝ → 𝑥 < +∞)
1615ad2antlr 473 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < +∞)
17 breq2 3824 . . . . . . . . . . . . . . . . . . 19 (𝑧 = +∞ → (𝑥 < 𝑧𝑥 < +∞))
1817adantl 271 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑥 < +∞))
1916, 18mpbird 165 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < 𝑧)
2019orcd 685 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑧 < 𝑦))
2120a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
22 mnflt 9185 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ → -∞ < 𝑦)
2322ad2antrr 472 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → -∞ < 𝑦)
24 breq1 3823 . . . . . . . . . . . . . . . . . . 19 (𝑧 = -∞ → (𝑧 < 𝑦 ↔ -∞ < 𝑦))
2524adantl 271 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑧 < 𝑦 ↔ -∞ < 𝑦))
2623, 25mpbird 165 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
2726olcd 686 . . . . . . . . . . . . . . . 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 1240 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
309, 29sylan2b 281 . . . . . . . . . . . . 13 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
3130anasss 391 . . . . . . . . . . . 12 ((𝑦 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
3231ancoms 264 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
33 ltpnf 9183 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℝ → 𝑧 < +∞)
3433adantl 271 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 < +∞)
35 breq2 3824 . . . . . . . . . . . . . . . . . . 19 (𝑦 = +∞ → (𝑧 < 𝑦𝑧 < +∞))
3635ad2antrr 472 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑧 < 𝑦𝑧 < +∞))
3734, 36mpbird 165 . . . . . . . . . . . . . . . . 17 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 < 𝑦)
3837olcd 686 . . . . . . . . . . . . . . . 16 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑧𝑧 < 𝑦))
3938a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
4015ad2antlr 473 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < +∞)
4117adantl 271 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑥 < +∞))
4240, 41mpbird 165 . . . . . . . . . . . . . . . . 17 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < 𝑧)
4342orcd 685 . . . . . . . . . . . . . . . 16 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑧 < 𝑦))
4443a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
45 mnfltpnf 9187 . . . . . . . . . . . . . . . . . . 19 -∞ < +∞
46 breq12 3825 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 = -∞ ∧ 𝑦 = +∞) → (𝑧 < 𝑦 ↔ -∞ < +∞))
4746ancoms 264 . . . . . . . . . . . . . . . . . . 19 ((𝑦 = +∞ ∧ 𝑧 = -∞) → (𝑧 < 𝑦 ↔ -∞ < +∞))
4845, 47mpbiri 166 . . . . . . . . . . . . . . . . . 18 ((𝑦 = +∞ ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
4948adantlr 461 . . . . . . . . . . . . . . . . 17 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
5049olcd 686 . . . . . . . . . . . . . . . 16 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑥 < 𝑧𝑧 < 𝑦))
5150a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
5239, 44, 513jaodan 1240 . . . . . . . . . . . . . 14 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
539, 52sylan2b 281 . . . . . . . . . . . . 13 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
5453anasss 391 . . . . . . . . . . . 12 ((𝑦 = +∞ ∧ (𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
5554ancoms 264 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
56 rexr 7477 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
57 nltmnf 9190 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
5856, 57syl 14 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → ¬ 𝑥 < -∞)
5958ad2antrr 472 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → ¬ 𝑥 < -∞)
60 breq2 3824 . . . . . . . . . . . . . 14 (𝑦 = -∞ → (𝑥 < 𝑦𝑥 < -∞))
6160adantl 271 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → (𝑥 < 𝑦𝑥 < -∞))
6259, 61mtbird 631 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → ¬ 𝑥 < 𝑦)
6362pm2.21d 582 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
6432, 55, 633jaodan 1240 . . . . . . . . . 10 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
658, 64sylan2b 281 . . . . . . . . 9 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
6665anasss 391 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ (𝑧 ∈ ℝ*𝑦 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
6766ancoms 264 . . . . . . 7 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
68 pnfnlt 9189 . . . . . . . . . 10 (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
6968ad2antlr 473 . . . . . . . . 9 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → ¬ +∞ < 𝑦)
70 breq1 3823 . . . . . . . . . 10 (𝑥 = +∞ → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
7170adantl 271 . . . . . . . . 9 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
7269, 71mtbird 631 . . . . . . . 8 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → ¬ 𝑥 < 𝑦)
7372pm2.21d 582 . . . . . . 7 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
74 df-3or 923 . . . . . . . . . . 11 ((𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞) ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞))
759, 74bitri 182 . . . . . . . . . 10 (𝑧 ∈ ℝ* ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞))
76 mnfltxr 9188 . . . . . . . . . . . . . . 15 ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) → -∞ < 𝑧)
7776adantl 271 . . . . . . . . . . . . . 14 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → -∞ < 𝑧)
78 breq1 3823 . . . . . . . . . . . . . . 15 (𝑥 = -∞ → (𝑥 < 𝑧 ↔ -∞ < 𝑧))
7978adantr 270 . . . . . . . . . . . . . 14 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑧 ↔ -∞ < 𝑧))
8077, 79mpbird 165 . . . . . . . . . . . . 13 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → 𝑥 < 𝑧)
8180orcd 685 . . . . . . . . . . . 12 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑧𝑧 < 𝑦))
8281a1d 22 . . . . . . . . . . 11 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
83 eqtr3 2104 . . . . . . . . . . . . 13 ((𝑥 = -∞ ∧ 𝑧 = -∞) → 𝑥 = 𝑧)
8483breq1d 3830 . . . . . . . . . . . 12 ((𝑥 = -∞ ∧ 𝑧 = -∞) → (𝑥 < 𝑦𝑧 < 𝑦))
85 olc 665 . . . . . . . . . . . 12 (𝑧 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦))
8684, 85syl6bi 161 . . . . . . . . . . 11 ((𝑥 = -∞ ∧ 𝑧 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
8782, 86jaodan 744 . . . . . . . . . 10 ((𝑥 = -∞ ∧ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
8875, 87sylan2b 281 . . . . . . . . 9 ((𝑥 = -∞ ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
8988ancoms 264 . . . . . . . 8 ((𝑧 ∈ ℝ*𝑥 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
9089adantlr 461 . . . . . . 7 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
9167, 73, 903jaodan 1240 . . . . . 6 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
92913impa 1136 . . . . 5 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
937, 92syl3an3b 1210 . . . 4 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ*𝑥 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
94933com13 1146 . . 3 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
9594rgen3 2456 . 2 𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ* (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦))
96 df-iso 4098 . 2 ( < Or ℝ* ↔ ( < Po ℝ* ∧ ∀𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ* (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦))))
976, 95, 96mpbir2an 886 1 < Or ℝ*
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  w3o 921  w3a 922   = wceq 1287  wtru 1288  wcel 1436  wral 2355   class class class wbr 3820   Po wpo 4095   Or wor 4096  cr 7293  +∞cpnf 7463  -∞cmnf 7464  *cxr 7465   < clt 7466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-cnex 7380  ax-resscn 7381  ax-pre-ltirr 7401  ax-pre-ltwlin 7402  ax-pre-lttrn 7403
This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-po 4097  df-iso 4098  df-xp 4417  df-pnf 7468  df-mnf 7469  df-xr 7470  df-ltxr 7471
This theorem is referenced by:  xrlelttr  9203  xrltletr  9204  xrletr  9205
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