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Theorem ltxrlt 11329
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 brun 5199 . . . . 5 (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴({-∞} × ℝ)𝐵))
2 brxp 5738 . . . . . . 7 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
3 elsni 4648 . . . . . . . 8 (𝐵 ∈ {+∞} → 𝐵 = +∞)
4 pnfnre 11300 . . . . . . . . . 10 +∞ ∉ ℝ
54neli 3046 . . . . . . . . 9 ¬ +∞ ∈ ℝ
6 simpr 484 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
7 eleq1 2827 . . . . . . . . . 10 (𝐵 = +∞ → (𝐵 ∈ ℝ ↔ +∞ ∈ ℝ))
86, 7imbitrid 244 . . . . . . . . 9 (𝐵 = +∞ → ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → +∞ ∈ ℝ))
95, 8mtoi 199 . . . . . . . 8 (𝐵 = +∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
103, 9syl 17 . . . . . . 7 (𝐵 ∈ {+∞} → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
112, 10simplbiim 504 . . . . . 6 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
12 brxp 5738 . . . . . . 7 (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
13 elsni 4648 . . . . . . . . 9 (𝐴 ∈ {-∞} → 𝐴 = -∞)
14 mnfnre 11302 . . . . . . . . . . 11 -∞ ∉ ℝ
1514neli 3046 . . . . . . . . . 10 ¬ -∞ ∈ ℝ
16 simpl 482 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ)
17 eleq1 2827 . . . . . . . . . . 11 (𝐴 = -∞ → (𝐴 ∈ ℝ ↔ -∞ ∈ ℝ))
1816, 17imbitrid 244 . . . . . . . . . 10 (𝐴 = -∞ → ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -∞ ∈ ℝ))
1915, 18mtoi 199 . . . . . . . . 9 (𝐴 = -∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
2013, 19syl 17 . . . . . . . 8 (𝐴 ∈ {-∞} → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
2120adantr 480 . . . . . . 7 ((𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
2212, 21sylbi 217 . . . . . 6 (𝐴({-∞} × ℝ)𝐵 → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
2311, 22jaoi 857 . . . . 5 ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴({-∞} × ℝ)𝐵) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
241, 23sylbi 217 . . . 4 (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
2524con2i 139 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵)
26 df-ltxr 11298 . . . . . . 7 < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2726equncomi 4170 . . . . . 6 < = ((((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)})
2827breqi 5154 . . . . 5 (𝐴 < 𝐵𝐴((((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)})𝐵)
29 brun 5199 . . . . 5 (𝐴((((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)})𝐵 ↔ (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵))
30 df-or 848 . . . . 5 ((𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵) ↔ (¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵))
3128, 29, 303bitri 297 . . . 4 (𝐴 < 𝐵 ↔ (¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵))
32 biimt 360 . . . 4 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵 ↔ (¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵)))
3331, 32bitr4id 290 . . 3 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → (𝐴 < 𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵))
3425, 33syl 17 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵))
35 breq12 5153 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 < 𝑦𝐴 < 𝐵))
36 df-3an 1088 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦))
3736opabbii 5215 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦)}
3835, 37brab2a 5782 . . 3 (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 < 𝐵))
3938baibr 536 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵))
4034, 39bitr4d 282 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  cun 3961  {csn 4631   class class class wbr 5148  {copab 5210   × cxp 5687  cr 11152   < cltrr 11157  +∞cpnf 11290  -∞cmnf 11291   < clt 11293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-ltxr 11298
This theorem is referenced by:  axlttri  11330  axlttrn  11331  axltadd  11332  axmulgt0  11333  axsup  11334
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