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Theorem xrltso 8788
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 8772 . . . . 5 (𝑥 ∈ ℝ* → ¬ 𝑥 < 𝑥)
21adantl 266 . . . 4 ((⊤ ∧ 𝑥 ∈ ℝ*) → ¬ 𝑥 < 𝑥)
3 xrlttr 8787 . . . . 5 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
43adantl 266 . . . 4 ((⊤ ∧ (𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*)) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
52, 4ispod 4066 . . 3 (⊤ → < Po ℝ*)
65trud 1266 . 2 < Po ℝ*
7 elxr 8767 . . . . 5 (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞))
8 elxr 8767 . . . . . . . . . 10 (𝑦 ∈ ℝ* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
9 elxr 8767 . . . . . . . . . . . . . 14 (𝑧 ∈ ℝ* ↔ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞))
10 simplr 490 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑥 ∈ ℝ)
11 simpll 489 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ)
12 simpr 107 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
13 axltwlin 7116 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
1410, 11, 12, 13syl3anc 1144 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
15 ltpnf 8773 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℝ → 𝑥 < +∞)
1615ad2antlr 466 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < +∞)
17 breq2 3793 . . . . . . . . . . . . . . . . . . 19 (𝑧 = +∞ → (𝑥 < 𝑧𝑥 < +∞))
1817adantl 266 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑥 < +∞))
1916, 18mpbird 160 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < 𝑧)
2019orcd 660 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑧 < 𝑦))
2120a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
22 mnflt 8775 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ → -∞ < 𝑦)
2322ad2antrr 465 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → -∞ < 𝑦)
24 breq1 3792 . . . . . . . . . . . . . . . . . . 19 (𝑧 = -∞ → (𝑧 < 𝑦 ↔ -∞ < 𝑦))
2524adantl 266 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑧 < 𝑦 ↔ -∞ < 𝑦))
2623, 25mpbird 160 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
2726olcd 661 . . . . . . . . . . . . . . . 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 1210 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
309, 29sylan2b 275 . . . . . . . . . . . . 13 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
3130anasss 385 . . . . . . . . . . . 12 ((𝑦 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
3231ancoms 259 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
33 ltpnf 8773 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℝ → 𝑧 < +∞)
3433adantl 266 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 < +∞)
35 breq2 3793 . . . . . . . . . . . . . . . . . . 19 (𝑦 = +∞ → (𝑧 < 𝑦𝑧 < +∞))
3635ad2antrr 465 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑧 < 𝑦𝑧 < +∞))
3734, 36mpbird 160 . . . . . . . . . . . . . . . . 17 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 < 𝑦)
3837olcd 661 . . . . . . . . . . . . . . . 16 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑧𝑧 < 𝑦))
3938a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
4015ad2antlr 466 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < +∞)
4117adantl 266 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑥 < +∞))
4240, 41mpbird 160 . . . . . . . . . . . . . . . . 17 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < 𝑧)
4342orcd 660 . . . . . . . . . . . . . . . 16 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑧 < 𝑦))
4443a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
45 mnfltpnf 8777 . . . . . . . . . . . . . . . . . . 19 -∞ < +∞
46 breq12 3794 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 = -∞ ∧ 𝑦 = +∞) → (𝑧 < 𝑦 ↔ -∞ < +∞))
4746ancoms 259 . . . . . . . . . . . . . . . . . . 19 ((𝑦 = +∞ ∧ 𝑧 = -∞) → (𝑧 < 𝑦 ↔ -∞ < +∞))
4845, 47mpbiri 161 . . . . . . . . . . . . . . . . . 18 ((𝑦 = +∞ ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
4948adantlr 454 . . . . . . . . . . . . . . . . 17 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
5049olcd 661 . . . . . . . . . . . . . . . 16 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑥 < 𝑧𝑧 < 𝑦))
5150a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
5239, 44, 513jaodan 1210 . . . . . . . . . . . . . 14 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
539, 52sylan2b 275 . . . . . . . . . . . . 13 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
5453anasss 385 . . . . . . . . . . . 12 ((𝑦 = +∞ ∧ (𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
5554ancoms 259 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
56 rexr 7100 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
57 nltmnf 8780 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
5856, 57syl 14 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → ¬ 𝑥 < -∞)
5958ad2antrr 465 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → ¬ 𝑥 < -∞)
60 breq2 3793 . . . . . . . . . . . . . 14 (𝑦 = -∞ → (𝑥 < 𝑦𝑥 < -∞))
6160adantl 266 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → (𝑥 < 𝑦𝑥 < -∞))
6259, 61mtbird 606 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → ¬ 𝑥 < 𝑦)
6362pm2.21d 557 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
6432, 55, 633jaodan 1210 . . . . . . . . . 10 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
658, 64sylan2b 275 . . . . . . . . 9 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
6665anasss 385 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ (𝑧 ∈ ℝ*𝑦 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
6766ancoms 259 . . . . . . 7 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
68 pnfnlt 8779 . . . . . . . . . 10 (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
6968ad2antlr 466 . . . . . . . . 9 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → ¬ +∞ < 𝑦)
70 breq1 3792 . . . . . . . . . 10 (𝑥 = +∞ → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
7170adantl 266 . . . . . . . . 9 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
7269, 71mtbird 606 . . . . . . . 8 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → ¬ 𝑥 < 𝑦)
7372pm2.21d 557 . . . . . . 7 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
74 df-3or 895 . . . . . . . . . . 11 ((𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞) ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞))
759, 74bitri 177 . . . . . . . . . 10 (𝑧 ∈ ℝ* ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞))
76 mnfltxr 8778 . . . . . . . . . . . . . . 15 ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) → -∞ < 𝑧)
7776adantl 266 . . . . . . . . . . . . . 14 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → -∞ < 𝑧)
78 breq1 3792 . . . . . . . . . . . . . . 15 (𝑥 = -∞ → (𝑥 < 𝑧 ↔ -∞ < 𝑧))
7978adantr 265 . . . . . . . . . . . . . 14 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑧 ↔ -∞ < 𝑧))
8077, 79mpbird 160 . . . . . . . . . . . . 13 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → 𝑥 < 𝑧)
8180orcd 660 . . . . . . . . . . . 12 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑧𝑧 < 𝑦))
8281a1d 22 . . . . . . . . . . 11 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
83 eqtr3 2073 . . . . . . . . . . . . 13 ((𝑥 = -∞ ∧ 𝑧 = -∞) → 𝑥 = 𝑧)
8483breq1d 3799 . . . . . . . . . . . 12 ((𝑥 = -∞ ∧ 𝑧 = -∞) → (𝑥 < 𝑦𝑧 < 𝑦))
85 olc 640 . . . . . . . . . . . 12 (𝑧 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦))
8684, 85syl6bi 156 . . . . . . . . . . 11 ((𝑥 = -∞ ∧ 𝑧 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
8782, 86jaodan 719 . . . . . . . . . 10 ((𝑥 = -∞ ∧ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
8875, 87sylan2b 275 . . . . . . . . 9 ((𝑥 = -∞ ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
8988ancoms 259 . . . . . . . 8 ((𝑧 ∈ ℝ*𝑥 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
9089adantlr 454 . . . . . . 7 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
9167, 73, 903jaodan 1210 . . . . . 6 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
92913impa 1108 . . . . 5 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
937, 92syl3an3b 1182 . . . 4 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ*𝑥 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
94933com13 1118 . . 3 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
9594rgen3 2421 . 2 𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ* (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦))
96 df-iso 4059 . 2 ( < Or ℝ* ↔ ( < Po ℝ* ∧ ∀𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ* (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦))))
976, 95, 96mpbir2an 858 1 < Or ℝ*
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 637  w3o 893  w3a 894   = wceq 1257  wtru 1258  wcel 1407  wral 2321   class class class wbr 3789   Po wpo 4056   Or wor 4057  cr 6916  +∞cpnf 7086  -∞cmnf 7087  *cxr 7088   < clt 7089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-cnex 7003  ax-resscn 7004  ax-pre-ltirr 7024  ax-pre-ltwlin 7025  ax-pre-lttrn 7026
This theorem depends on definitions:  df-bi 114  df-3or 895  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-nel 2313  df-ral 2326  df-rex 2327  df-rab 2330  df-v 2574  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-br 3790  df-opab 3844  df-po 4058  df-iso 4059  df-xp 4376  df-pnf 7091  df-mnf 7092  df-xr 7093  df-ltxr 7094
This theorem is referenced by:  xrlelttr  8793  xrltletr  8794  xrletr  8795
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