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Theorem xrltso 9589
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 9573 . . . . 5 (𝑥 ∈ ℝ* → ¬ 𝑥 < 𝑥)
21adantl 275 . . . 4 ((⊤ ∧ 𝑥 ∈ ℝ*) → ¬ 𝑥 < 𝑥)
3 xrlttr 9588 . . . . 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 4226 . . 3 (⊤ → < Po ℝ*)
65mptru 1340 . 2 < Po ℝ*
7 elxr 9570 . . . . 5 (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞))
8 elxr 9570 . . . . . . . . . 10 (𝑦 ∈ ℝ* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
9 elxr 9570 . . . . . . . . . . . . . 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 7839 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
1410, 11, 12, 13syl3anc 1216 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
15 ltpnf 9574 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℝ → 𝑥 < +∞)
1615ad2antlr 480 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < +∞)
17 breq2 3933 . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑧 < 𝑦))
2120a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
22 mnflt 9576 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ → -∞ < 𝑦)
2322ad2antrr 479 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → -∞ < 𝑦)
24 breq1 3932 . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . 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 1284 . . . . . . . . . . . . . 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 9574 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℝ → 𝑧 < +∞)
3433adantl 275 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 < +∞)
35 breq2 3933 . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . 16 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑧 < 𝑦))
4443a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
45 mnfltpnf 9578 . . . . . . . . . . . . . . . . . . 19 -∞ < +∞
46 breq12 3934 . . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . 16 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑥 < 𝑧𝑧 < 𝑦))
5150a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
5239, 44, 513jaodan 1284 . . . . . . . . . . . . . 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 7818 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
57 nltmnf 9581 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
5856, 57syl 14 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → ¬ 𝑥 < -∞)
5958ad2antrr 479 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → ¬ 𝑥 < -∞)
60 breq2 3933 . . . . . . . . . . . . . 14 (𝑦 = -∞ → (𝑥 < 𝑦𝑥 < -∞))
6160adantl 275 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → (𝑥 < 𝑦𝑥 < -∞))
6259, 61mtbird 662 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → ¬ 𝑥 < 𝑦)
6362pm2.21d 608 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
6432, 55, 633jaodan 1284 . . . . . . . . . 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 9580 . . . . . . . . . 10 (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
6968ad2antlr 480 . . . . . . . . 9 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → ¬ +∞ < 𝑦)
70 breq1 3932 . . . . . . . . . 10 (𝑥 = +∞ → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
7170adantl 275 . . . . . . . . 9 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
7269, 71mtbird 662 . . . . . . . 8 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → ¬ 𝑥 < 𝑦)
7372pm2.21d 608 . . . . . . 7 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
74 df-3or 963 . . . . . . . . . . 11 ((𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞) ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞))
759, 74bitri 183 . . . . . . . . . 10 (𝑧 ∈ ℝ* ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞))
76 mnfltxr 9579 . . . . . . . . . . . . . . 15 ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) → -∞ < 𝑧)
7776adantl 275 . . . . . . . . . . . . . 14 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → -∞ < 𝑧)
78 breq1 3932 . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . 12 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑧𝑧 < 𝑦))
8281a1d 22 . . . . . . . . . . 11 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
83 eqtr3 2159 . . . . . . . . . . . . 13 ((𝑥 = -∞ ∧ 𝑧 = -∞) → 𝑥 = 𝑧)
8483breq1d 3939 . . . . . . . . . . . 12 ((𝑥 = -∞ ∧ 𝑧 = -∞) → (𝑥 < 𝑦𝑧 < 𝑦))
85 olc 700 . . . . . . . . . . . 12 (𝑧 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦))
8684, 85syl6bi 162 . . . . . . . . . . 11 ((𝑥 = -∞ ∧ 𝑧 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
8782, 86jaodan 786 . . . . . . . . . 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 1284 . . . . . 6 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
92913impa 1176 . . . . 5 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
937, 92syl3an3b 1254 . . . 4 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ*𝑥 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
94933com13 1186 . . 3 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
9594rgen3 2519 . 2 𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ* (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦))
96 df-iso 4219 . 2 ( < Or ℝ* ↔ ( < Po ℝ* ∧ ∀𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ* (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦))))
976, 95, 96mpbir2an 926 1 < Or ℝ*
Colors of variables: wff set class
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  cr 7626  +∞cpnf 7804  -∞cmnf 7805  *cxr 7806   < clt 7807
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 7718  ax-resscn 7719  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741
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 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812
This theorem is referenced by:  xrlelttr  9596  xrltletr  9597  xrletr  9598  xrmaxiflemlub  11024
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