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

Theorem ltxrlt 7455
Description: The standard less-than < and the extended real less-than < are identical when restricted to the non-extended reals . (Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
ltxrlt ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))

Proof of Theorem ltxrlt
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ltxr 7430 . . . . 5 < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
21breqi 3817 . . . 4 (𝐴 < 𝐵𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵)
3 brun 3857 . . . 4 (𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵 ↔ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
42, 3bitri 182 . . 3 (𝐴 < 𝐵 ↔ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
5 eleq1 2145 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ∈ ℝ ↔ 𝐴 ∈ ℝ))
6 breq1 3814 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 < 𝑦𝐴 < 𝑦))
75, 63anbi13d 1246 . . . . . 6 (𝑥 = 𝐴 → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 < 𝑦)))
8 eleq1 2145 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 ∈ ℝ ↔ 𝐵 ∈ ℝ))
9 breq2 3815 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 < 𝑦𝐴 < 𝐵))
108, 93anbi23d 1247 . . . . . 6 (𝑦 = 𝐵 → ((𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 < 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
11 eqid 2083 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}
127, 10, 11brabg 4060 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵 ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
13 simp3 941 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
1412, 13syl6bi 161 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴 < 𝐵))
15 brun 3857 . . . . 5 (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴({-∞} × ℝ)𝐵))
16 brxp 4431 . . . . . . . . . . 11 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
1716simprbi 269 . . . . . . . . . 10 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐵 ∈ {+∞})
18 elsni 3440 . . . . . . . . . 10 (𝐵 ∈ {+∞} → 𝐵 = +∞)
1917, 18syl 14 . . . . . . . . 9 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐵 = +∞)
2019a1i 9 . . . . . . . 8 (𝐵 ∈ ℝ → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐵 = +∞))
21 renepnf 7438 . . . . . . . . 9 (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
2221neneqd 2270 . . . . . . . 8 (𝐵 ∈ ℝ → ¬ 𝐵 = +∞)
23 pm2.24 584 . . . . . . . 8 (𝐵 = +∞ → (¬ 𝐵 = +∞ → 𝐴 < 𝐵))
2420, 22, 23syl6ci 1375 . . . . . . 7 (𝐵 ∈ ℝ → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴 < 𝐵))
2524adantl 271 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴 < 𝐵))
26 brxp 4431 . . . . . . . . . . 11 (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
2726simplbi 268 . . . . . . . . . 10 (𝐴({-∞} × ℝ)𝐵𝐴 ∈ {-∞})
28 elsni 3440 . . . . . . . . . 10 (𝐴 ∈ {-∞} → 𝐴 = -∞)
2927, 28syl 14 . . . . . . . . 9 (𝐴({-∞} × ℝ)𝐵𝐴 = -∞)
3029a1i 9 . . . . . . . 8 (𝐴 ∈ ℝ → (𝐴({-∞} × ℝ)𝐵𝐴 = -∞))
31 renemnf 7439 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
3231neneqd 2270 . . . . . . . 8 (𝐴 ∈ ℝ → ¬ 𝐴 = -∞)
33 pm2.24 584 . . . . . . . 8 (𝐴 = -∞ → (¬ 𝐴 = -∞ → 𝐴 < 𝐵))
3430, 32, 33syl6ci 1375 . . . . . . 7 (𝐴 ∈ ℝ → (𝐴({-∞} × ℝ)𝐵𝐴 < 𝐵))
3534adantr 270 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴({-∞} × ℝ)𝐵𝐴 < 𝐵))
3625, 35jaod 670 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴({-∞} × ℝ)𝐵) → 𝐴 < 𝐵))
3715, 36syl5bi 150 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵𝐴 < 𝐵))
3814, 37jaod 670 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵) → 𝐴 < 𝐵))
394, 38syl5bi 150 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
40123adant3 959 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵 ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
4140ibir 175 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵)
4241orcd 685 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
4342, 4sylibr 132 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
44433expia 1141 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
4539, 44impbid 127 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  w3a 920   = wceq 1285  wcel 1434  cun 2982  {csn 3422   class class class wbr 3811  {copab 3864   × cxp 4399  cr 7252   < cltrr 7257  +∞cpnf 7422  -∞cmnf 7423   < clt 7425
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-cnex 7339  ax-resscn 7340
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-xp 4407  df-pnf 7427  df-mnf 7428  df-ltxr 7430
This theorem is referenced by:  axltirr  7456  axltwlin  7457  axlttrn  7458  axltadd  7459  axapti  7460  axmulgt0  7461  0lt1  7513  recexre  7955  recexgt0  7957  remulext1  7976  arch  8562  caucvgrelemcau  10240  caucvgre  10241
  Copyright terms: Public domain W3C validator