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Theorem ltxrlt 7531
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 7506 . . . . 5 < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
21breqi 3843 . . . 4 (𝐴 < 𝐵𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵)
3 brun 3883 . . . 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 2150 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ∈ ℝ ↔ 𝐴 ∈ ℝ))
6 breq1 3840 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 < 𝑦𝐴 < 𝑦))
75, 63anbi13d 1250 . . . . . 6 (𝑥 = 𝐴 → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 < 𝑦)))
8 eleq1 2150 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 ∈ ℝ ↔ 𝐵 ∈ ℝ))
9 breq2 3841 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 < 𝑦𝐴 < 𝐵))
108, 93anbi23d 1251 . . . . . 6 (𝑦 = 𝐵 → ((𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 < 𝑦) ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
11 eqid 2088 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}
127, 10, 11brabg 4087 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵 ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
13 simp3 945 . . . . 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 3883 . . . . 5 (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴({-∞} × ℝ)𝐵))
16 brxp 4458 . . . . . . . . . . 11 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
1716simprbi 269 . . . . . . . . . 10 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐵 ∈ {+∞})
18 elsni 3459 . . . . . . . . . 10 (𝐵 ∈ {+∞} → 𝐵 = +∞)
1917, 18syl 14 . . . . . . . . 9 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐵 = +∞)
2019a1i 9 . . . . . . . 8 (𝐵 ∈ ℝ → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐵 = +∞))
21 renepnf 7514 . . . . . . . . 9 (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
2221neneqd 2276 . . . . . . . 8 (𝐵 ∈ ℝ → ¬ 𝐵 = +∞)
23 pm2.24 586 . . . . . . . 8 (𝐵 = +∞ → (¬ 𝐵 = +∞ → 𝐴 < 𝐵))
2420, 22, 23syl6ci 1379 . . . . . . 7 (𝐵 ∈ ℝ → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴 < 𝐵))
2524adantl 271 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴 < 𝐵))
26 brxp 4458 . . . . . . . . . . 11 (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
2726simplbi 268 . . . . . . . . . 10 (𝐴({-∞} × ℝ)𝐵𝐴 ∈ {-∞})
28 elsni 3459 . . . . . . . . . 10 (𝐴 ∈ {-∞} → 𝐴 = -∞)
2927, 28syl 14 . . . . . . . . 9 (𝐴({-∞} × ℝ)𝐵𝐴 = -∞)
3029a1i 9 . . . . . . . 8 (𝐴 ∈ ℝ → (𝐴({-∞} × ℝ)𝐵𝐴 = -∞))
31 renemnf 7515 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
3231neneqd 2276 . . . . . . . 8 (𝐴 ∈ ℝ → ¬ 𝐴 = -∞)
33 pm2.24 586 . . . . . . . 8 (𝐴 = -∞ → (¬ 𝐴 = -∞ → 𝐴 < 𝐵))
3430, 32, 33syl6ci 1379 . . . . . . 7 (𝐴 ∈ ℝ → (𝐴({-∞} × ℝ)𝐵𝐴 < 𝐵))
3534adantr 270 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴({-∞} × ℝ)𝐵𝐴 < 𝐵))
3625, 35jaod 672 . . . . 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 672 . . 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 963 . . . . . 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 687 . . . 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 1145 . 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 664  w3a 924   = wceq 1289  wcel 1438  cun 2995  {csn 3441   class class class wbr 3837  {copab 3890   × cxp 4426  cr 7328   < cltrr 7333  +∞cpnf 7498  -∞cmnf 7499   < clt 7501
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-xp 4434  df-pnf 7503  df-mnf 7504  df-ltxr 7506
This theorem is referenced by:  axltirr  7532  axltwlin  7533  axlttrn  7534  axltadd  7535  axapti  7536  axmulgt0  7537  0lt1  7589  recexre  8031  recexgt0  8033  remulext1  8052  arch  8640  caucvgrelemcau  10378  caucvgre  10379
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