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Theorem cvrval 37777
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 31266 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 37776 . . . . 5 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))})
5 3anass 1096 . . . . . 6 (((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)) ↔ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))))
65opabbii 5173 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}
74, 6eqtrdi 2789 . . . 4 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))})
87breqd 5117 . . 3 (𝐾𝐴 → (𝑋𝐶𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌))
983ad2ant1 1134 . 2 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌))
10 df-br 5107 . . . 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 5109 . . . . . 6 (𝑥 = 𝑋 → (𝑥 < 𝑦𝑋 < 𝑦))
12 breq1 5109 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 < 𝑧𝑋 < 𝑧))
1312anbi1d 631 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑥 < 𝑧𝑧 < 𝑦) ↔ (𝑋 < 𝑧𝑧 < 𝑦)))
1413rexbidv 3172 . . . . . . 7 (𝑥 = 𝑋 → (∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦) ↔ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦)))
1514notbid 318 . . . . . 6 (𝑥 = 𝑋 → (¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦) ↔ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦)))
1611, 15anbi12d 632 . . . . 5 (𝑥 = 𝑋 → ((𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)) ↔ (𝑋 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦))))
17 breq2 5110 . . . . . 6 (𝑦 = 𝑌 → (𝑋 < 𝑦𝑋 < 𝑌))
18 breq2 5110 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑧 < 𝑦𝑧 < 𝑌))
1918anbi2d 630 . . . . . . . 8 (𝑦 = 𝑌 → ((𝑋 < 𝑧𝑧 < 𝑦) ↔ (𝑋 < 𝑧𝑧 < 𝑌)))
2019rexbidv 3172 . . . . . . 7 (𝑦 = 𝑌 → (∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦) ↔ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)))
2120notbid 318 . . . . . 6 (𝑦 = 𝑌 → (¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦) ↔ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)))
2217, 21anbi12d 632 . . . . 5 (𝑦 = 𝑌 → ((𝑋 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦)) ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
2316, 22opelopab2 5499 . . . 4 ((𝑋𝐵𝑌𝐵) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))} ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
2410, 23bitrid 283 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
25243adant1 1131 . 2 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
269, 25bitrd 279 1 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wrex 3070  cop 4593   class class class wbr 5106  {copab 5168  cfv 6497  Basecbs 17088  ltcplt 18202  ccvr 37770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-covers 37774
This theorem is referenced by:  cvrlt  37778  cvrnbtwn  37779  cvrval2  37782  cvrcon3b  37785  lautcvr  38601
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