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Theorem ltxrlt 7083
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 7063 . . . . 5 < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
21breqi 3770 . . . 4 (𝐴 < 𝐵𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵)
3 brun 3810 . . . 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 173 . . 3 (𝐴 < 𝐵 ↔ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
5 eleq1 2100 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ∈ ℝ ↔ 𝐴 ∈ ℝ))
6 breq1 3767 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 < 𝑦𝐴 < 𝑦))
75, 63anbi13d 1209 . . . . . 6 (𝑥 = 𝐴 → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 < 𝑦)))
8 eleq1 2100 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 ∈ ℝ ↔ 𝐵 ∈ ℝ))
9 breq2 3768 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 < 𝑦𝐴 < 𝐵))
108, 93anbi23d 1210 . . . . . 6 (𝑦 = 𝐵 → ((𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 < 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
11 eqid 2040 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}
127, 10, 11brabg 4006 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵 ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
13 simp3 906 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
1412, 13syl6bi 152 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴 < 𝐵))
15 brun 3810 . . . . 5 (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴({-∞} × ℝ)𝐵))
16 brxp 4375 . . . . . . . . . . 11 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
1716simprbi 260 . . . . . . . . . 10 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐵 ∈ {+∞})
18 elsni 3393 . . . . . . . . . 10 (𝐵 ∈ {+∞} → 𝐵 = +∞)
1917, 18syl 14 . . . . . . . . 9 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐵 = +∞)
2019a1i 9 . . . . . . . 8 (𝐵 ∈ ℝ → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐵 = +∞))
21 renepnf 7071 . . . . . . . . 9 (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
2221neneqd 2226 . . . . . . . 8 (𝐵 ∈ ℝ → ¬ 𝐵 = +∞)
23 pm2.24 551 . . . . . . . 8 (𝐵 = +∞ → (¬ 𝐵 = +∞ → 𝐴 < 𝐵))
2420, 22, 23syl6ci 1334 . . . . . . 7 (𝐵 ∈ ℝ → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴 < 𝐵))
2524adantl 262 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴 < 𝐵))
26 brxp 4375 . . . . . . . . . . 11 (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
2726simplbi 259 . . . . . . . . . 10 (𝐴({-∞} × ℝ)𝐵𝐴 ∈ {-∞})
28 elsni 3393 . . . . . . . . . 10 (𝐴 ∈ {-∞} → 𝐴 = -∞)
2927, 28syl 14 . . . . . . . . 9 (𝐴({-∞} × ℝ)𝐵𝐴 = -∞)
3029a1i 9 . . . . . . . 8 (𝐴 ∈ ℝ → (𝐴({-∞} × ℝ)𝐵𝐴 = -∞))
31 renemnf 7072 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
3231neneqd 2226 . . . . . . . 8 (𝐴 ∈ ℝ → ¬ 𝐴 = -∞)
33 pm2.24 551 . . . . . . . 8 (𝐴 = -∞ → (¬ 𝐴 = -∞ → 𝐴 < 𝐵))
3430, 32, 33syl6ci 1334 . . . . . . 7 (𝐴 ∈ ℝ → (𝐴({-∞} × ℝ)𝐵𝐴 < 𝐵))
3534adantr 261 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴({-∞} × ℝ)𝐵𝐴 < 𝐵))
3625, 35jaod 637 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴({-∞} × ℝ)𝐵) → 𝐴 < 𝐵))
3715, 36syl5bi 141 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵𝐴 < 𝐵))
3814, 37jaod 637 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵) → 𝐴 < 𝐵))
394, 38syl5bi 141 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
40123adant3 924 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵 ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
4140ibir 166 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵)
4241orcd 652 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
4342, 4sylibr 137 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
44433expia 1106 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
4539, 44impbid 120 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wb 98  wo 629  w3a 885   = wceq 1243  wcel 1393  cun 2915  {csn 3375   class class class wbr 3764  {copab 3817   × cxp 4343  cr 6886   < cltrr 6891  +∞cpnf 7055  -∞cmnf 7056   < clt 7058
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6973  ax-resscn 6974
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-pnf 7060  df-mnf 7061  df-ltxr 7063
This theorem is referenced by:  axltirr  7084  axltwlin  7085  axlttrn  7086  axltadd  7087  axapti  7088  axmulgt0  7089  0lt1  7139  recexre  7567  recexgt0  7569  remulext1  7588  arch  8176  caucvgrelemcau  9553  caucvgre  9554
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