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Theorem xrltso 9783
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 9766 . . . . 5 (𝑥 ∈ ℝ* → ¬ 𝑥 < 𝑥)
21adantl 277 . . . 4 ((⊤ ∧ 𝑥 ∈ ℝ*) → ¬ 𝑥 < 𝑥)
3 xrlttr 9782 . . . . 5 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
43adantl 277 . . . 4 ((⊤ ∧ (𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*)) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
52, 4ispod 4301 . . 3 (⊤ → < Po ℝ*)
65mptru 1362 . 2 < Po ℝ*
7 elxr 9763 . . . . 5 (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞))
8 elxr 9763 . . . . . . . . . 10 (𝑦 ∈ ℝ* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
9 elxr 9763 . . . . . . . . . . . . . 14 (𝑧 ∈ ℝ* ↔ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞))
10 simplr 528 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑥 ∈ ℝ)
11 simpll 527 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ)
12 simpr 110 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
13 axltwlin 8015 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
1410, 11, 12, 13syl3anc 1238 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
15 ltpnf 9767 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℝ → 𝑥 < +∞)
1615ad2antlr 489 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < +∞)
17 breq2 4004 . . . . . . . . . . . . . . . . . . 19 (𝑧 = +∞ → (𝑥 < 𝑧𝑥 < +∞))
1817adantl 277 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑥 < +∞))
1916, 18mpbird 167 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < 𝑧)
2019orcd 733 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑧 < 𝑦))
2120a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
22 mnflt 9770 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ → -∞ < 𝑦)
2322ad2antrr 488 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → -∞ < 𝑦)
24 breq1 4003 . . . . . . . . . . . . . . . . . . 19 (𝑧 = -∞ → (𝑧 < 𝑦 ↔ -∞ < 𝑦))
2524adantl 277 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑧 < 𝑦 ↔ -∞ < 𝑦))
2623, 25mpbird 167 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
2726olcd 734 . . . . . . . . . . . . . . . 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 1306 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
309, 29sylan2b 287 . . . . . . . . . . . . 13 (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
3130anasss 399 . . . . . . . . . . . 12 ((𝑦 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
3231ancoms 268 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
33 ltpnf 9767 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℝ → 𝑧 < +∞)
3433adantl 277 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 < +∞)
35 breq2 4004 . . . . . . . . . . . . . . . . . . 19 (𝑦 = +∞ → (𝑧 < 𝑦𝑧 < +∞))
3635ad2antrr 488 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑧 < 𝑦𝑧 < +∞))
3734, 36mpbird 167 . . . . . . . . . . . . . . . . 17 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → 𝑧 < 𝑦)
3837olcd 734 . . . . . . . . . . . . . . . 16 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑧𝑧 < 𝑦))
3938a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
4015ad2antlr 489 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < +∞)
4117adantl 277 . . . . . . . . . . . . . . . . . 18 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑥 < +∞))
4240, 41mpbird 167 . . . . . . . . . . . . . . . . 17 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → 𝑥 < 𝑧)
4342orcd 733 . . . . . . . . . . . . . . . 16 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑧𝑧 < 𝑦))
4443a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
45 mnfltpnf 9772 . . . . . . . . . . . . . . . . . . 19 -∞ < +∞
46 breq12 4005 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 = -∞ ∧ 𝑦 = +∞) → (𝑧 < 𝑦 ↔ -∞ < +∞))
4746ancoms 268 . . . . . . . . . . . . . . . . . . 19 ((𝑦 = +∞ ∧ 𝑧 = -∞) → (𝑧 < 𝑦 ↔ -∞ < +∞))
4845, 47mpbiri 168 . . . . . . . . . . . . . . . . . 18 ((𝑦 = +∞ ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
4948adantlr 477 . . . . . . . . . . . . . . . . 17 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → 𝑧 < 𝑦)
5049olcd 734 . . . . . . . . . . . . . . . 16 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑥 < 𝑧𝑧 < 𝑦))
5150a1d 22 . . . . . . . . . . . . . . 15 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
5239, 44, 513jaodan 1306 . . . . . . . . . . . . . 14 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
539, 52sylan2b 287 . . . . . . . . . . . . 13 (((𝑦 = +∞ ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
5453anasss 399 . . . . . . . . . . . 12 ((𝑦 = +∞ ∧ (𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
5554ancoms 268 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
56 rexr 7993 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
57 nltmnf 9775 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
5856, 57syl 14 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → ¬ 𝑥 < -∞)
5958ad2antrr 488 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → ¬ 𝑥 < -∞)
60 breq2 4004 . . . . . . . . . . . . . 14 (𝑦 = -∞ → (𝑥 < 𝑦𝑥 < -∞))
6160adantl 277 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → (𝑥 < 𝑦𝑥 < -∞))
6259, 61mtbird 673 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → ¬ 𝑥 < 𝑦)
6362pm2.21d 619 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
6432, 55, 633jaodan 1306 . . . . . . . . . 10 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
658, 64sylan2b 287 . . . . . . . . 9 (((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ*) ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
6665anasss 399 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ (𝑧 ∈ ℝ*𝑦 ∈ ℝ*)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
6766ancoms 268 . . . . . . 7 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
68 pnfnlt 9774 . . . . . . . . . 10 (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
6968ad2antlr 489 . . . . . . . . 9 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → ¬ +∞ < 𝑦)
70 breq1 4003 . . . . . . . . . 10 (𝑥 = +∞ → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
7170adantl 277 . . . . . . . . 9 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
7269, 71mtbird 673 . . . . . . . 8 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → ¬ 𝑥 < 𝑦)
7372pm2.21d 619 . . . . . . 7 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
74 df-3or 979 . . . . . . . . . . 11 ((𝑧 ∈ ℝ ∨ 𝑧 = +∞ ∨ 𝑧 = -∞) ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞))
759, 74bitri 184 . . . . . . . . . 10 (𝑧 ∈ ℝ* ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞))
76 mnfltxr 9773 . . . . . . . . . . . . . . 15 ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) → -∞ < 𝑧)
7776adantl 277 . . . . . . . . . . . . . 14 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → -∞ < 𝑧)
78 breq1 4003 . . . . . . . . . . . . . . 15 (𝑥 = -∞ → (𝑥 < 𝑧 ↔ -∞ < 𝑧))
7978adantr 276 . . . . . . . . . . . . . 14 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑧 ↔ -∞ < 𝑧))
8077, 79mpbird 167 . . . . . . . . . . . . 13 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → 𝑥 < 𝑧)
8180orcd 733 . . . . . . . . . . . 12 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑧𝑧 < 𝑦))
8281a1d 22 . . . . . . . . . . 11 ((𝑥 = -∞ ∧ (𝑧 ∈ ℝ ∨ 𝑧 = +∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
83 eqtr3 2197 . . . . . . . . . . . . 13 ((𝑥 = -∞ ∧ 𝑧 = -∞) → 𝑥 = 𝑧)
8483breq1d 4010 . . . . . . . . . . . 12 ((𝑥 = -∞ ∧ 𝑧 = -∞) → (𝑥 < 𝑦𝑧 < 𝑦))
85 olc 711 . . . . . . . . . . . 12 (𝑧 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦))
8684, 85syl6bi 163 . . . . . . . . . . 11 ((𝑥 = -∞ ∧ 𝑧 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
8782, 86jaodan 797 . . . . . . . . . 10 ((𝑥 = -∞ ∧ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∨ 𝑧 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
8875, 87sylan2b 287 . . . . . . . . 9 ((𝑥 = -∞ ∧ 𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
8988ancoms 268 . . . . . . . 8 ((𝑧 ∈ ℝ*𝑥 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
9089adantlr 477 . . . . . . 7 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = -∞) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
9167, 73, 903jaodan 1306 . . . . . 6 (((𝑧 ∈ ℝ*𝑦 ∈ ℝ*) ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
92913impa 1194 . . . . 5 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
937, 92syl3an3b 1276 . . . 4 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ*𝑥 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
94933com13 1208 . . 3 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
9594rgen3 2564 . 2 𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ* (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦))
96 df-iso 4294 . 2 ( < Or ℝ* ↔ ( < Po ℝ* ∧ ∀𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ* (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦))))
976, 95, 96mpbir2an 942 1 < Or ℝ*
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  w3o 977  w3a 978   = wceq 1353  wtru 1354  wcel 2148  wral 2455   class class class wbr 4000   Po wpo 4291   Or wor 4292  cr 7801  +∞cpnf 7979  -∞cmnf 7980  *cxr 7981   < clt 7982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-po 4293  df-iso 4294  df-xp 4629  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987
This theorem is referenced by:  xrlelttr  9793  xrltletr  9794  xrletr  9795  xrmaxiflemlub  11240
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