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Theorem ltxrlt 10364
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 4862 . . . . 5 (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴({-∞} × ℝ)𝐵))
2 brxp 5325 . . . . . . 7 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
3 elsni 4353 . . . . . . . . 9 (𝐵 ∈ {+∞} → 𝐵 = +∞)
4 pnfnre 10337 . . . . . . . . . . 11 +∞ ∉ ℝ
54neli 3042 . . . . . . . . . 10 ¬ +∞ ∈ ℝ
6 simpr 477 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
7 eleq1 2832 . . . . . . . . . . 11 (𝐵 = +∞ → (𝐵 ∈ ℝ ↔ +∞ ∈ ℝ))
86, 7syl5ib 235 . . . . . . . . . 10 (𝐵 = +∞ → ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → +∞ ∈ ℝ))
95, 8mtoi 190 . . . . . . . . 9 (𝐵 = +∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
103, 9syl 17 . . . . . . . 8 (𝐵 ∈ {+∞} → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
1110adantl 473 . . . . . . 7 ((𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
122, 11sylbi 208 . . . . . 6 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
13 brxp 5325 . . . . . . 7 (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
14 elsni 4353 . . . . . . . . 9 (𝐴 ∈ {-∞} → 𝐴 = -∞)
15 mnfnre 10338 . . . . . . . . . . 11 -∞ ∉ ℝ
1615neli 3042 . . . . . . . . . 10 ¬ -∞ ∈ ℝ
17 simpl 474 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ)
18 eleq1 2832 . . . . . . . . . . 11 (𝐴 = -∞ → (𝐴 ∈ ℝ ↔ -∞ ∈ ℝ))
1917, 18syl5ib 235 . . . . . . . . . 10 (𝐴 = -∞ → ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -∞ ∈ ℝ))
2016, 19mtoi 190 . . . . . . . . 9 (𝐴 = -∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
2114, 20syl 17 . . . . . . . 8 (𝐴 ∈ {-∞} → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
2221adantr 472 . . . . . . 7 ((𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
2313, 22sylbi 208 . . . . . 6 (𝐴({-∞} × ℝ)𝐵 → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
2412, 23jaoi 883 . . . . 5 ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴({-∞} × ℝ)𝐵) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
251, 24sylbi 208 . . . 4 (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
2625con2i 136 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵)
27 biimt 351 . . . 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)))
28 df-ltxr 10335 . . . . . . 7 < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2928equncomi 3923 . . . . . 6 < = ((((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)})
3029breqi 4817 . . . . 5 (𝐴 < 𝐵𝐴((((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)})𝐵)
31 brun 4862 . . . . 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))
32 df-or 874 . . . . 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))
3330, 31, 323bitri 288 . . . 4 (𝐴 < 𝐵 ↔ (¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵))
3427, 33syl6rbbr 281 . . 3 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → (𝐴 < 𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵))
3526, 34syl 17 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵))
36 breq12 4816 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 < 𝑦𝐴 < 𝐵))
37 df-3an 1109 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦))
3837opabbii 4878 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦)}
3936, 38brab2a 5366 . . 3 (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 < 𝐵))
4039baibr 532 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵))
4135, 40bitr4d 273 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  cun 3732  {csn 4336   class class class wbr 4811  {copab 4873   × cxp 5277  cr 10190   < cltrr 10195  +∞cpnf 10327  -∞cmnf 10328   < clt 10330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-resscn 10248
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-er 7949  df-en 8163  df-dom 8164  df-sdom 8165  df-pnf 10332  df-mnf 10333  df-ltxr 10335
This theorem is referenced by:  axlttri  10365  axlttrn  10366  axltadd  10367  axmulgt0  10368  axsup  10369
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