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Theorem ltxrlt 8022
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 7996 . . . . 5 < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
21breqi 4009 . . . 4 (𝐴 < 𝐵𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵)
3 brun 4054 . . . 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 184 . . 3 (𝐴 < 𝐵 ↔ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
5 eleq1 2240 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ∈ ℝ ↔ 𝐴 ∈ ℝ))
6 breq1 4006 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 < 𝑦𝐴 < 𝑦))
75, 63anbi13d 1314 . . . . . 6 (𝑥 = 𝐴 → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 < 𝑦)))
8 eleq1 2240 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 ∈ ℝ ↔ 𝐵 ∈ ℝ))
9 breq2 4007 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 < 𝑦𝐴 < 𝐵))
108, 93anbi23d 1315 . . . . . 6 (𝑦 = 𝐵 → ((𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 < 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
11 eqid 2177 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}
127, 10, 11brabg 4269 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵 ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
13 simp3 999 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
1412, 13syl6bi 163 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴 < 𝐵))
15 brun 4054 . . . . 5 (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴({-∞} × ℝ)𝐵))
16 brxp 4657 . . . . . . . . . . 11 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
1716simprbi 275 . . . . . . . . . 10 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐵 ∈ {+∞})
18 elsni 3610 . . . . . . . . . 10 (𝐵 ∈ {+∞} → 𝐵 = +∞)
1917, 18syl 14 . . . . . . . . 9 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐵 = +∞)
2019a1i 9 . . . . . . . 8 (𝐵 ∈ ℝ → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐵 = +∞))
21 renepnf 8004 . . . . . . . . 9 (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
2221neneqd 2368 . . . . . . . 8 (𝐵 ∈ ℝ → ¬ 𝐵 = +∞)
23 pm2.24 621 . . . . . . . 8 (𝐵 = +∞ → (¬ 𝐵 = +∞ → 𝐴 < 𝐵))
2420, 22, 23syl6ci 1445 . . . . . . 7 (𝐵 ∈ ℝ → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴 < 𝐵))
2524adantl 277 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴 < 𝐵))
26 brxp 4657 . . . . . . . . . . 11 (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
2726simplbi 274 . . . . . . . . . 10 (𝐴({-∞} × ℝ)𝐵𝐴 ∈ {-∞})
28 elsni 3610 . . . . . . . . . 10 (𝐴 ∈ {-∞} → 𝐴 = -∞)
2927, 28syl 14 . . . . . . . . 9 (𝐴({-∞} × ℝ)𝐵𝐴 = -∞)
3029a1i 9 . . . . . . . 8 (𝐴 ∈ ℝ → (𝐴({-∞} × ℝ)𝐵𝐴 = -∞))
31 renemnf 8005 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
3231neneqd 2368 . . . . . . . 8 (𝐴 ∈ ℝ → ¬ 𝐴 = -∞)
33 pm2.24 621 . . . . . . . 8 (𝐴 = -∞ → (¬ 𝐴 = -∞ → 𝐴 < 𝐵))
3430, 32, 33syl6ci 1445 . . . . . . 7 (𝐴 ∈ ℝ → (𝐴({-∞} × ℝ)𝐵𝐴 < 𝐵))
3534adantr 276 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴({-∞} × ℝ)𝐵𝐴 < 𝐵))
3625, 35jaod 717 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴({-∞} × ℝ)𝐵) → 𝐴 < 𝐵))
3715, 36biimtrid 152 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵𝐴 < 𝐵))
3814, 37jaod 717 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵) → 𝐴 < 𝐵))
394, 38biimtrid 152 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
40123adant3 1017 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵 ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
4140ibir 177 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵)
4241orcd 733 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
4342, 4sylibr 134 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
44433expia 1205 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
4539, 44impbid 129 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  w3a 978   = wceq 1353  wcel 2148  cun 3127  {csn 3592   class class class wbr 4003  {copab 4063   × cxp 4624  cr 7809   < cltrr 7814  +∞cpnf 7988  -∞cmnf 7989   < clt 7991
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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902
This theorem depends on definitions:  df-bi 117  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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-pnf 7993  df-mnf 7994  df-ltxr 7996
This theorem is referenced by:  axltirr  8023  axltwlin  8024  axlttrn  8025  axltadd  8026  axapti  8027  axmulgt0  8028  axsuploc  8029  0lt1  8083  recexre  8534  recexgt0  8536  remulext1  8555  arch  9172  caucvgrelemcau  10988  caucvgre  10989
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