ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xrlttri3 Structured version   GIF version

Theorem xrlttri3 8468
Description: Extended real version of lttri3 6875. (Contributed by NM, 9-Feb-2006.)
Assertion
Ref Expression
xrlttri3 ((A * B *) → (A = B ↔ (¬ A < B ¬ B < A)))

Proof of Theorem xrlttri3
StepHypRef Expression
1 elxr 8446 . 2 (A * ↔ (A A = +∞ A = -∞))
2 elxr 8446 . 2 (B * ↔ (B B = +∞ B = -∞))
3 lttri3 6875 . . . . . 6 ((A B ℝ) → (A = B ↔ (¬ A < B ¬ B < A)))
43ancoms 255 . . . . 5 ((B A ℝ) → (A = B ↔ (¬ A < B ¬ B < A)))
5 renepnf 6850 . . . . . . . . . 10 (B ℝ → B ≠ +∞)
65adantr 261 . . . . . . . . 9 ((B A = +∞) → B ≠ +∞)
7 neeq2 2214 . . . . . . . . . 10 (A = +∞ → (BAB ≠ +∞))
87adantl 262 . . . . . . . . 9 ((B A = +∞) → (BAB ≠ +∞))
96, 8mpbird 156 . . . . . . . 8 ((B A = +∞) → BA)
109necomd 2285 . . . . . . 7 ((B A = +∞) → AB)
1110neneqd 2221 . . . . . 6 ((B A = +∞) → ¬ A = B)
12 ltpnf 8452 . . . . . . . . 9 (B ℝ → B < +∞)
1312adantr 261 . . . . . . . 8 ((B A = +∞) → B < +∞)
14 breq2 3759 . . . . . . . . 9 (A = +∞ → (B < AB < +∞))
1514adantl 262 . . . . . . . 8 ((B A = +∞) → (B < AB < +∞))
1613, 15mpbird 156 . . . . . . 7 ((B A = +∞) → B < A)
17 notnot1 559 . . . . . . . . 9 ((A < B B < A) → ¬ ¬ (A < B B < A))
1817olcs 654 . . . . . . . 8 (B < A → ¬ ¬ (A < B B < A))
19 ioran 668 . . . . . . . 8 (¬ (A < B B < A) ↔ (¬ A < B ¬ B < A))
2018, 19sylnib 600 . . . . . . 7 (B < A → ¬ (¬ A < B ¬ B < A))
2116, 20syl 14 . . . . . 6 ((B A = +∞) → ¬ (¬ A < B ¬ B < A))
2211, 212falsed 617 . . . . 5 ((B A = +∞) → (A = B ↔ (¬ A < B ¬ B < A)))
23 renemnf 6851 . . . . . . . . . 10 (B ℝ → B ≠ -∞)
2423adantr 261 . . . . . . . . 9 ((B A = -∞) → B ≠ -∞)
25 neeq2 2214 . . . . . . . . . 10 (A = -∞ → (BAB ≠ -∞))
2625adantl 262 . . . . . . . . 9 ((B A = -∞) → (BAB ≠ -∞))
2724, 26mpbird 156 . . . . . . . 8 ((B A = -∞) → BA)
2827necomd 2285 . . . . . . 7 ((B A = -∞) → AB)
2928neneqd 2221 . . . . . 6 ((B A = -∞) → ¬ A = B)
30 mnflt 8454 . . . . . . . . 9 (B ℝ → -∞ < B)
3130adantr 261 . . . . . . . 8 ((B A = -∞) → -∞ < B)
32 breq1 3758 . . . . . . . . 9 (A = -∞ → (A < B ↔ -∞ < B))
3332adantl 262 . . . . . . . 8 ((B A = -∞) → (A < B ↔ -∞ < B))
3431, 33mpbird 156 . . . . . . 7 ((B A = -∞) → A < B)
35 orc 632 . . . . . . 7 (A < B → (A < B B < A))
36 oranim 806 . . . . . . 7 ((A < B B < A) → ¬ (¬ A < B ¬ B < A))
3734, 35, 363syl 17 . . . . . 6 ((B A = -∞) → ¬ (¬ A < B ¬ B < A))
3829, 372falsed 617 . . . . 5 ((B A = -∞) → (A = B ↔ (¬ A < B ¬ B < A)))
394, 22, 383jaodan 1200 . . . 4 ((B (A A = +∞ A = -∞)) → (A = B ↔ (¬ A < B ¬ B < A)))
4039ancoms 255 . . 3 (((A A = +∞ A = -∞) B ℝ) → (A = B ↔ (¬ A < B ¬ B < A)))
41 renepnf 6850 . . . . . . . . 9 (A ℝ → A ≠ +∞)
4241adantl 262 . . . . . . . 8 ((B = +∞ A ℝ) → A ≠ +∞)
43 neeq2 2214 . . . . . . . . 9 (B = +∞ → (ABA ≠ +∞))
4443adantr 261 . . . . . . . 8 ((B = +∞ A ℝ) → (ABA ≠ +∞))
4542, 44mpbird 156 . . . . . . 7 ((B = +∞ A ℝ) → AB)
4645neneqd 2221 . . . . . 6 ((B = +∞ A ℝ) → ¬ A = B)
47 ltpnf 8452 . . . . . . . . 9 (A ℝ → A < +∞)
4847adantl 262 . . . . . . . 8 ((B = +∞ A ℝ) → A < +∞)
49 breq2 3759 . . . . . . . . 9 (B = +∞ → (A < BA < +∞))
5049adantr 261 . . . . . . . 8 ((B = +∞ A ℝ) → (A < BA < +∞))
5148, 50mpbird 156 . . . . . . 7 ((B = +∞ A ℝ) → A < B)
5251, 35, 363syl 17 . . . . . 6 ((B = +∞ A ℝ) → ¬ (¬ A < B ¬ B < A))
5346, 522falsed 617 . . . . 5 ((B = +∞ A ℝ) → (A = B ↔ (¬ A < B ¬ B < A)))
54 eqtr3 2056 . . . . . . 7 ((B = +∞ A = +∞) → B = A)
5554eqcomd 2042 . . . . . 6 ((B = +∞ A = +∞) → A = B)
56 pnfxr 8442 . . . . . . . . 9 +∞ *
57 xrltnr 8451 . . . . . . . . 9 (+∞ * → ¬ +∞ < +∞)
5856, 57ax-mp 7 . . . . . . . 8 ¬ +∞ < +∞
59 breq12 3760 . . . . . . . . 9 ((A = +∞ B = +∞) → (A < B ↔ +∞ < +∞))
6059ancoms 255 . . . . . . . 8 ((B = +∞ A = +∞) → (A < B ↔ +∞ < +∞))
6158, 60mtbiri 599 . . . . . . 7 ((B = +∞ A = +∞) → ¬ A < B)
62 breq12 3760 . . . . . . . 8 ((B = +∞ A = +∞) → (B < A ↔ +∞ < +∞))
6358, 62mtbiri 599 . . . . . . 7 ((B = +∞ A = +∞) → ¬ B < A)
6461, 63jca 290 . . . . . 6 ((B = +∞ A = +∞) → (¬ A < B ¬ B < A))
6555, 642thd 164 . . . . 5 ((B = +∞ A = +∞) → (A = B ↔ (¬ A < B ¬ B < A)))
66 mnfnepnf 8448 . . . . . . . . 9 -∞ ≠ +∞
67 eqeq12 2049 . . . . . . . . . 10 ((A = -∞ B = +∞) → (A = B ↔ -∞ = +∞))
6867necon3bid 2240 . . . . . . . . 9 ((A = -∞ B = +∞) → (AB ↔ -∞ ≠ +∞))
6966, 68mpbiri 157 . . . . . . . 8 ((A = -∞ B = +∞) → AB)
7069ancoms 255 . . . . . . 7 ((B = +∞ A = -∞) → AB)
7170neneqd 2221 . . . . . 6 ((B = +∞ A = -∞) → ¬ A = B)
72 mnfltpnf 8456 . . . . . . . . 9 -∞ < +∞
73 breq12 3760 . . . . . . . . 9 ((A = -∞ B = +∞) → (A < B ↔ -∞ < +∞))
7472, 73mpbiri 157 . . . . . . . 8 ((A = -∞ B = +∞) → A < B)
7574ancoms 255 . . . . . . 7 ((B = +∞ A = -∞) → A < B)
7675, 35, 363syl 17 . . . . . 6 ((B = +∞ A = -∞) → ¬ (¬ A < B ¬ B < A))
7771, 762falsed 617 . . . . 5 ((B = +∞ A = -∞) → (A = B ↔ (¬ A < B ¬ B < A)))
7853, 65, 773jaodan 1200 . . . 4 ((B = +∞ (A A = +∞ A = -∞)) → (A = B ↔ (¬ A < B ¬ B < A)))
7978ancoms 255 . . 3 (((A A = +∞ A = -∞) B = +∞) → (A = B ↔ (¬ A < B ¬ B < A)))
80 renemnf 6851 . . . . . . . . 9 (A ℝ → A ≠ -∞)
8180adantl 262 . . . . . . . 8 ((B = -∞ A ℝ) → A ≠ -∞)
82 neeq2 2214 . . . . . . . . 9 (B = -∞ → (ABA ≠ -∞))
8382adantr 261 . . . . . . . 8 ((B = -∞ A ℝ) → (ABA ≠ -∞))
8481, 83mpbird 156 . . . . . . 7 ((B = -∞ A ℝ) → AB)
8584neneqd 2221 . . . . . 6 ((B = -∞ A ℝ) → ¬ A = B)
86 mnflt 8454 . . . . . . . . 9 (A ℝ → -∞ < A)
8786adantl 262 . . . . . . . 8 ((B = -∞ A ℝ) → -∞ < A)
88 breq1 3758 . . . . . . . . 9 (B = -∞ → (B < A ↔ -∞ < A))
8988adantr 261 . . . . . . . 8 ((B = -∞ A ℝ) → (B < A ↔ -∞ < A))
9087, 89mpbird 156 . . . . . . 7 ((B = -∞ A ℝ) → B < A)
9190, 20syl 14 . . . . . 6 ((B = -∞ A ℝ) → ¬ (¬ A < B ¬ B < A))
9285, 912falsed 617 . . . . 5 ((B = -∞ A ℝ) → (A = B ↔ (¬ A < B ¬ B < A)))
9366neii 2205 . . . . . . . . . 10 ¬ -∞ = +∞
94 eqeq12 2049 . . . . . . . . . 10 ((B = -∞ A = +∞) → (B = A ↔ -∞ = +∞))
9593, 94mtbiri 599 . . . . . . . . 9 ((B = -∞ A = +∞) → ¬ B = A)
9695neneqad 2278 . . . . . . . 8 ((B = -∞ A = +∞) → BA)
9796necomd 2285 . . . . . . 7 ((B = -∞ A = +∞) → AB)
9897neneqd 2221 . . . . . 6 ((B = -∞ A = +∞) → ¬ A = B)
99 breq12 3760 . . . . . . . 8 ((B = -∞ A = +∞) → (B < A ↔ -∞ < +∞))
10072, 99mpbiri 157 . . . . . . 7 ((B = -∞ A = +∞) → B < A)
101100, 20syl 14 . . . . . 6 ((B = -∞ A = +∞) → ¬ (¬ A < B ¬ B < A))
10298, 1012falsed 617 . . . . 5 ((B = -∞ A = +∞) → (A = B ↔ (¬ A < B ¬ B < A)))
103 eqtr3 2056 . . . . . . 7 ((A = -∞ B = -∞) → A = B)
104103ancoms 255 . . . . . 6 ((B = -∞ A = -∞) → A = B)
105 mnfxr 8444 . . . . . . . . 9 -∞ *
106 xrltnr 8451 . . . . . . . . 9 (-∞ * → ¬ -∞ < -∞)
107105, 106ax-mp 7 . . . . . . . 8 ¬ -∞ < -∞
108 breq12 3760 . . . . . . . . 9 ((A = -∞ B = -∞) → (A < B ↔ -∞ < -∞))
109108ancoms 255 . . . . . . . 8 ((B = -∞ A = -∞) → (A < B ↔ -∞ < -∞))
110107, 109mtbiri 599 . . . . . . 7 ((B = -∞ A = -∞) → ¬ A < B)
111 breq12 3760 . . . . . . . 8 ((B = -∞ A = -∞) → (B < A ↔ -∞ < -∞))
112107, 111mtbiri 599 . . . . . . 7 ((B = -∞ A = -∞) → ¬ B < A)
113110, 112jca 290 . . . . . 6 ((B = -∞ A = -∞) → (¬ A < B ¬ B < A))
114104, 1132thd 164 . . . . 5 ((B = -∞ A = -∞) → (A = B ↔ (¬ A < B ¬ B < A)))
11592, 102, 1143jaodan 1200 . . . 4 ((B = -∞ (A A = +∞ A = -∞)) → (A = B ↔ (¬ A < B ¬ B < A)))
116115ancoms 255 . . 3 (((A A = +∞ A = -∞) B = -∞) → (A = B ↔ (¬ A < B ¬ B < A)))
11740, 79, 1163jaodan 1200 . 2 (((A A = +∞ A = -∞) (B B = +∞ B = -∞)) → (A = B ↔ (¬ A < B ¬ B < A)))
1181, 2, 117syl2anb 275 1 ((A * B *) → (A = B ↔ (¬ A < B ¬ B < A)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628   w3o 883   = wceq 1242   wcel 1390  wne 2201   class class class wbr 3755  cr 6690  +∞cpnf 6834  -∞cmnf 6835  *cxr 6836   < clt 6837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6754  ax-resscn 6755  ax-pre-ltirr 6775  ax-pre-apti 6778
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-pnf 6839  df-mnf 6840  df-xr 6841  df-ltxr 6842
This theorem is referenced by:  xrletri3  8471  iccid  8544
  Copyright terms: Public domain W3C validator