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Theorem ltxrlt 11225
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 5156 . . . . 5 (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴({-∞} × ℝ)𝐵))
2 brxp 5681 . . . . . . 7 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
3 elsni 4603 . . . . . . . 8 (𝐵 ∈ {+∞} → 𝐵 = +∞)
4 pnfnre 11196 . . . . . . . . . 10 +∞ ∉ ℝ
54neli 3051 . . . . . . . . 9 ¬ +∞ ∈ ℝ
6 simpr 485 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
7 eleq1 2825 . . . . . . . . . 10 (𝐵 = +∞ → (𝐵 ∈ ℝ ↔ +∞ ∈ ℝ))
86, 7imbitrid 243 . . . . . . . . 9 (𝐵 = +∞ → ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → +∞ ∈ ℝ))
95, 8mtoi 198 . . . . . . . 8 (𝐵 = +∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
103, 9syl 17 . . . . . . 7 (𝐵 ∈ {+∞} → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
112, 10simplbiim 505 . . . . . 6 (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
12 brxp 5681 . . . . . . 7 (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
13 elsni 4603 . . . . . . . . 9 (𝐴 ∈ {-∞} → 𝐴 = -∞)
14 mnfnre 11198 . . . . . . . . . . 11 -∞ ∉ ℝ
1514neli 3051 . . . . . . . . . 10 ¬ -∞ ∈ ℝ
16 simpl 483 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ)
17 eleq1 2825 . . . . . . . . . . 11 (𝐴 = -∞ → (𝐴 ∈ ℝ ↔ -∞ ∈ ℝ))
1816, 17imbitrid 243 . . . . . . . . . 10 (𝐴 = -∞ → ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -∞ ∈ ℝ))
1915, 18mtoi 198 . . . . . . . . 9 (𝐴 = -∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
2013, 19syl 17 . . . . . . . 8 (𝐴 ∈ {-∞} → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
2120adantr 481 . . . . . . 7 ((𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
2212, 21sylbi 216 . . . . . 6 (𝐴({-∞} × ℝ)𝐵 → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
2311, 22jaoi 855 . . . . 5 ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵𝐴({-∞} × ℝ)𝐵) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
241, 23sylbi 216 . . . 4 (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
2524con2i 139 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵)
26 df-ltxr 11194 . . . . . . 7 < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2726equncomi 4115 . . . . . 6 < = ((((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)})
2827breqi 5111 . . . . 5 (𝐴 < 𝐵𝐴((((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)})𝐵)
29 brun 5156 . . . . 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 846 . . . . 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 296 . . . 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 289 . . 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 5110 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 < 𝑦𝐴 < 𝐵))
36 df-3an 1089 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦))
3736opabbii 5172 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 < 𝑦)}
3835, 37brab2a 5725 . . 3 (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 < 𝐵))
3938baibr 537 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)}𝐵))
4034, 39bitr4d 281 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  cun 3908  {csn 4586   class class class wbr 5105  {copab 5167   × cxp 5631  cr 11050   < cltrr 11055  +∞cpnf 11186  -∞cmnf 11187   < clt 11189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-resscn 11108
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-ltxr 11194
This theorem is referenced by:  axlttri  11226  axlttrn  11227  axltadd  11228  axmulgt0  11229  axsup  11230
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