Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvrval Structured version   Visualization version   GIF version

Theorem cvrval 34874
Description: Binary relation expressing 𝐵 covers 𝐴, which means that 𝐵 is larger than 𝐴 and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (cvbr 29269 analog.) (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
cvrfval.b 𝐵 = (Base‘𝐾)
cvrfval.s < = (lt‘𝐾)
cvrfval.c 𝐶 = ( ⋖ ‘𝐾)
Assertion
Ref Expression
cvrval ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
Distinct variable groups:   𝑧,𝐵   𝑧,𝐾   𝑧,𝑋   𝑧,𝑌
Allowed substitution hints:   𝐴(𝑧)   𝐶(𝑧)   < (𝑧)

Proof of Theorem cvrval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvrfval.b . . . . . 6 𝐵 = (Base‘𝐾)
2 cvrfval.s . . . . . 6 < = (lt‘𝐾)
3 cvrfval.c . . . . . 6 𝐶 = ( ⋖ ‘𝐾)
41, 2, 3cvrfval 34873 . . . . 5 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))})
5 3anass 1059 . . . . . 6 (((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)) ↔ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))))
65opabbii 4750 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}
74, 6syl6eq 2701 . . . 4 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))})
87breqd 4696 . . 3 (𝐾𝐴 → (𝑋𝐶𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌))
983ad2ant1 1102 . 2 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌))
10 df-br 4686 . . . 4 (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))})
11 breq1 4688 . . . . . 6 (𝑥 = 𝑋 → (𝑥 < 𝑦𝑋 < 𝑦))
12 breq1 4688 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 < 𝑧𝑋 < 𝑧))
1312anbi1d 741 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑥 < 𝑧𝑧 < 𝑦) ↔ (𝑋 < 𝑧𝑧 < 𝑦)))
1413rexbidv 3081 . . . . . . 7 (𝑥 = 𝑋 → (∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦) ↔ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦)))
1514notbid 307 . . . . . 6 (𝑥 = 𝑋 → (¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦) ↔ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦)))
1611, 15anbi12d 747 . . . . 5 (𝑥 = 𝑋 → ((𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)) ↔ (𝑋 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦))))
17 breq2 4689 . . . . . 6 (𝑦 = 𝑌 → (𝑋 < 𝑦𝑋 < 𝑌))
18 breq2 4689 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑧 < 𝑦𝑧 < 𝑌))
1918anbi2d 740 . . . . . . . 8 (𝑦 = 𝑌 → ((𝑋 < 𝑧𝑧 < 𝑦) ↔ (𝑋 < 𝑧𝑧 < 𝑌)))
2019rexbidv 3081 . . . . . . 7 (𝑦 = 𝑌 → (∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦) ↔ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)))
2120notbid 307 . . . . . 6 (𝑦 = 𝑌 → (¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦) ↔ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)))
2217, 21anbi12d 747 . . . . 5 (𝑦 = 𝑌 → ((𝑋 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦)) ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
2316, 22opelopab2 5025 . . . 4 ((𝑋𝐵𝑌𝐵) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))} ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
2410, 23syl5bb 272 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
25243adant1 1099 . 2 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
269, 25bitrd 268 1 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942  cop 4216   class class class wbr 4685  {copab 4745  cfv 5926  Basecbs 15904  ltcplt 16988  ccvr 34867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-covers 34871
This theorem is referenced by:  cvrlt  34875  cvrnbtwn  34876  cvrval2  34879  cvrcon3b  34882  lautcvr  35696
  Copyright terms: Public domain W3C validator