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Theorem ltxrlt 6862
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 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 6842 . . . . 5 < = ({⟨x, y⟩ ∣ (x y x < y)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
21breqi 3761 . . . 4 (A < BA({⟨x, y⟩ ∣ (x y x < y)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))B)
3 brun 3801 . . . 4 (A({⟨x, y⟩ ∣ (x y x < y)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))B ↔ (A{⟨x, y⟩ ∣ (x y x < y)}B A(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))B))
42, 3bitri 173 . . 3 (A < B ↔ (A{⟨x, y⟩ ∣ (x y x < y)}B A(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))B))
5 eleq1 2097 . . . . . . 7 (x = A → (x ℝ ↔ A ℝ))
6 breq1 3758 . . . . . . 7 (x = A → (x < yA < y))
75, 63anbi13d 1208 . . . . . 6 (x = A → ((x y x < y) ↔ (A y A < y)))
8 eleq1 2097 . . . . . . 7 (y = B → (y ℝ ↔ B ℝ))
9 breq2 3759 . . . . . . 7 (y = B → (A < yA < B))
108, 93anbi23d 1209 . . . . . 6 (y = B → ((A y A < y) ↔ (A B A < B)))
11 eqid 2037 . . . . . 6 {⟨x, y⟩ ∣ (x y x < y)} = {⟨x, y⟩ ∣ (x y x < y)}
127, 10, 11brabg 3997 . . . . 5 ((A B ℝ) → (A{⟨x, y⟩ ∣ (x y x < y)}B ↔ (A B A < B)))
13 simp3 905 . . . . 5 ((A B A < B) → A < B)
1412, 13syl6bi 152 . . . 4 ((A B ℝ) → (A{⟨x, y⟩ ∣ (x y x < y)}BA < B))
15 brun 3801 . . . . 5 (A(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))B ↔ (A((ℝ ∪ {-∞}) × {+∞})B A({-∞} × ℝ)B))
16 brxp 4318 . . . . . . . . . . 11 (A((ℝ ∪ {-∞}) × {+∞})B ↔ (A (ℝ ∪ {-∞}) B {+∞}))
1716simprbi 260 . . . . . . . . . 10 (A((ℝ ∪ {-∞}) × {+∞})BB {+∞})
18 elsni 3391 . . . . . . . . . 10 (B {+∞} → B = +∞)
1917, 18syl 14 . . . . . . . . 9 (A((ℝ ∪ {-∞}) × {+∞})BB = +∞)
2019a1i 9 . . . . . . . 8 (B ℝ → (A((ℝ ∪ {-∞}) × {+∞})BB = +∞))
21 renepnf 6850 . . . . . . . . 9 (B ℝ → B ≠ +∞)
2221neneqd 2221 . . . . . . . 8 (B ℝ → ¬ B = +∞)
23 pm2.24 551 . . . . . . . 8 (B = +∞ → (¬ B = +∞ → A < B))
2420, 22, 23syl6ci 1331 . . . . . . 7 (B ℝ → (A((ℝ ∪ {-∞}) × {+∞})BA < B))
2524adantl 262 . . . . . 6 ((A B ℝ) → (A((ℝ ∪ {-∞}) × {+∞})BA < B))
26 brxp 4318 . . . . . . . . . . 11 (A({-∞} × ℝ)B ↔ (A {-∞} B ℝ))
2726simplbi 259 . . . . . . . . . 10 (A({-∞} × ℝ)BA {-∞})
28 elsni 3391 . . . . . . . . . 10 (A {-∞} → A = -∞)
2927, 28syl 14 . . . . . . . . 9 (A({-∞} × ℝ)BA = -∞)
3029a1i 9 . . . . . . . 8 (A ℝ → (A({-∞} × ℝ)BA = -∞))
31 renemnf 6851 . . . . . . . . 9 (A ℝ → A ≠ -∞)
3231neneqd 2221 . . . . . . . 8 (A ℝ → ¬ A = -∞)
33 pm2.24 551 . . . . . . . 8 (A = -∞ → (¬ A = -∞ → A < B))
3430, 32, 33syl6ci 1331 . . . . . . 7 (A ℝ → (A({-∞} × ℝ)BA < B))
3534adantr 261 . . . . . 6 ((A B ℝ) → (A({-∞} × ℝ)BA < B))
3625, 35jaod 636 . . . . 5 ((A B ℝ) → ((A((ℝ ∪ {-∞}) × {+∞})B A({-∞} × ℝ)B) → A < B))
3715, 36syl5bi 141 . . . 4 ((A B ℝ) → (A(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))BA < B))
3814, 37jaod 636 . . 3 ((A B ℝ) → ((A{⟨x, y⟩ ∣ (x y x < y)}B A(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))B) → A < B))
394, 38syl5bi 141 . 2 ((A B ℝ) → (A < BA < B))
40123adant3 923 . . . . . 6 ((A B A < B) → (A{⟨x, y⟩ ∣ (x y x < y)}B ↔ (A B A < B)))
4140ibir 166 . . . . 5 ((A B A < B) → A{⟨x, y⟩ ∣ (x y x < y)}B)
4241orcd 651 . . . 4 ((A B A < B) → (A{⟨x, y⟩ ∣ (x y x < y)}B A(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))B))
4342, 4sylibr 137 . . 3 ((A B A < B) → A < B)
44433expia 1105 . 2 ((A B ℝ) → (A < BA < B))
4539, 44impbid 120 1 ((A B ℝ) → (A < BA < B))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628   w3a 884   = wceq 1242   wcel 1390  cun 2909  {csn 3367   class class class wbr 3755  {copab 3808   × cxp 4286  cr 6690   < cltrr 6695  +∞cpnf 6834  -∞cmnf 6835   < clt 6837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6754  ax-resscn 6755
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-pnf 6839  df-mnf 6840  df-ltxr 6842
This theorem is referenced by:  axltirr  6863  axltwlin  6864  axlttrn  6865  axltadd  6866  axapti  6867  axmulgt0  6868  0lt1  6918  recexre  7342  recexgt0  7344  remulext1  7363  arch  7934
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