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Theorem ltxrlt 7143
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 7123 . . . . 5 < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
21breqi 3797 . . . 4 (𝐴 < 𝐵𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵)
3 brun 3837 . . . 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 177 . . 3 (𝐴 < 𝐵 ↔ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
5 eleq1 2116 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ∈ ℝ ↔ 𝐴 ∈ ℝ))
6 breq1 3794 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 < 𝑦𝐴 < 𝑦))
75, 63anbi13d 1220 . . . . . 6 (𝑥 = 𝐴 → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 < 𝑦)))
8 eleq1 2116 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 ∈ ℝ ↔ 𝐵 ∈ ℝ))
9 breq2 3795 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 < 𝑦𝐴 < 𝐵))
108, 93anbi23d 1221 . . . . . 6 (𝑦 = 𝐵 → ((𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 < 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
11 eqid 2056 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}
127, 10, 11brabg 4033 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵 ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
13 simp3 917 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
1412, 13syl6bi 156 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴 < 𝐵))
15 brun 3837 . . . . 5 (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴({-∞} × ℝ)𝐵))
16 brxp 4402 . . . . . . . . . . 11 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
1716simprbi 264 . . . . . . . . . 10 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐵 ∈ {+∞})
18 elsni 3420 . . . . . . . . . 10 (𝐵 ∈ {+∞} → 𝐵 = +∞)
1917, 18syl 14 . . . . . . . . 9 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐵 = +∞)
2019a1i 9 . . . . . . . 8 (𝐵 ∈ ℝ → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐵 = +∞))
21 renepnf 7131 . . . . . . . . 9 (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
2221neneqd 2241 . . . . . . . 8 (𝐵 ∈ ℝ → ¬ 𝐵 = +∞)
23 pm2.24 561 . . . . . . . 8 (𝐵 = +∞ → (¬ 𝐵 = +∞ → 𝐴 < 𝐵))
2420, 22, 23syl6ci 1350 . . . . . . 7 (𝐵 ∈ ℝ → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴 < 𝐵))
2524adantl 266 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴 < 𝐵))
26 brxp 4402 . . . . . . . . . . 11 (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
2726simplbi 263 . . . . . . . . . 10 (𝐴({-∞} × ℝ)𝐵𝐴 ∈ {-∞})
28 elsni 3420 . . . . . . . . . 10 (𝐴 ∈ {-∞} → 𝐴 = -∞)
2927, 28syl 14 . . . . . . . . 9 (𝐴({-∞} × ℝ)𝐵𝐴 = -∞)
3029a1i 9 . . . . . . . 8 (𝐴 ∈ ℝ → (𝐴({-∞} × ℝ)𝐵𝐴 = -∞))
31 renemnf 7132 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
3231neneqd 2241 . . . . . . . 8 (𝐴 ∈ ℝ → ¬ 𝐴 = -∞)
33 pm2.24 561 . . . . . . . 8 (𝐴 = -∞ → (¬ 𝐴 = -∞ → 𝐴 < 𝐵))
3430, 32, 33syl6ci 1350 . . . . . . 7 (𝐴 ∈ ℝ → (𝐴({-∞} × ℝ)𝐵𝐴 < 𝐵))
3534adantr 265 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴({-∞} × ℝ)𝐵𝐴 < 𝐵))
3625, 35jaod 647 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴({-∞} × ℝ)𝐵) → 𝐴 < 𝐵))
3715, 36syl5bi 145 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵𝐴 < 𝐵))
3814, 37jaod 647 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵) → 𝐴 < 𝐵))
394, 38syl5bi 145 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
40123adant3 935 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵 ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
4140ibir 170 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵)
4241orcd 662 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
4342, 4sylibr 141 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
44433expia 1117 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
4539, 44impbid 124 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639  w3a 896   = wceq 1259  wcel 1409  cun 2942  {csn 3402   class class class wbr 3791  {copab 3844   × cxp 4370  cr 6945   < cltrr 6950  +∞cpnf 7115  -∞cmnf 7116   < clt 7118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-cnex 7032  ax-resscn 7033
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-xp 4378  df-pnf 7120  df-mnf 7121  df-ltxr 7123
This theorem is referenced by:  axltirr  7144  axltwlin  7145  axlttrn  7146  axltadd  7147  axapti  7148  axmulgt0  7149  0lt1  7201  recexre  7642  recexgt0  7644  remulext1  7663  arch  8235  caucvgrelemcau  9806  caucvgre  9807
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