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Theorem cvrval 38134
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 31530 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 38133 . . . . 5 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))})
5 3anass 1095 . . . . . 6 (((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)) ↔ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))))
65opabbii 5215 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}
74, 6eqtrdi 2788 . . . 4 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))})
87breqd 5159 . . 3 (𝐾𝐴 → (𝑋𝐶𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌))
983ad2ant1 1133 . 2 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌))
10 df-br 5149 . . . 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 5151 . . . . . 6 (𝑥 = 𝑋 → (𝑥 < 𝑦𝑋 < 𝑦))
12 breq1 5151 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 < 𝑧𝑋 < 𝑧))
1312anbi1d 630 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑥 < 𝑧𝑧 < 𝑦) ↔ (𝑋 < 𝑧𝑧 < 𝑦)))
1413rexbidv 3178 . . . . . . 7 (𝑥 = 𝑋 → (∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦) ↔ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦)))
1514notbid 317 . . . . . 6 (𝑥 = 𝑋 → (¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦) ↔ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦)))
1611, 15anbi12d 631 . . . . 5 (𝑥 = 𝑋 → ((𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)) ↔ (𝑋 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦))))
17 breq2 5152 . . . . . 6 (𝑦 = 𝑌 → (𝑋 < 𝑦𝑋 < 𝑌))
18 breq2 5152 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑧 < 𝑦𝑧 < 𝑌))
1918anbi2d 629 . . . . . . . 8 (𝑦 = 𝑌 → ((𝑋 < 𝑧𝑧 < 𝑦) ↔ (𝑋 < 𝑧𝑧 < 𝑌)))
2019rexbidv 3178 . . . . . . 7 (𝑦 = 𝑌 → (∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦) ↔ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)))
2120notbid 317 . . . . . 6 (𝑦 = 𝑌 → (¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦) ↔ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)))
2217, 21anbi12d 631 . . . . 5 (𝑦 = 𝑌 → ((𝑋 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦)) ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
2316, 22opelopab2 5541 . . . 4 ((𝑋𝐵𝑌𝐵) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))} ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
2410, 23bitrid 282 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
25243adant1 1130 . 2 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
269, 25bitrd 278 1 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3070  cop 4634   class class class wbr 5148  {copab 5210  cfv 6543  Basecbs 17143  ltcplt 18260  ccvr 38127
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-covers 38131
This theorem is referenced by:  cvrlt  38135  cvrnbtwn  38136  cvrval2  38139  cvrcon3b  38142  lautcvr  38958
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