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Theorem cvrval 38187
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 31566 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 38186 . . . . 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 5216 . . . . 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 5160 . . 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 5150 . . . 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 5152 . . . . . 6 (𝑥 = 𝑋 → (𝑥 < 𝑦𝑋 < 𝑦))
12 breq1 5152 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 < 𝑧𝑋 < 𝑧))
1312anbi1d 631 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑥 < 𝑧𝑧 < 𝑦) ↔ (𝑋 < 𝑧𝑧 < 𝑦)))
1413rexbidv 3179 . . . . . . 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 5153 . . . . . 6 (𝑦 = 𝑌 → (𝑋 < 𝑦𝑋 < 𝑌))
18 breq2 5153 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑧 < 𝑦𝑧 < 𝑌))
1918anbi2d 630 . . . . . . . 8 (𝑦 = 𝑌 → ((𝑋 < 𝑧𝑧 < 𝑦) ↔ (𝑋 < 𝑧𝑧 < 𝑌)))
2019rexbidv 3179 . . . . . . 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 5542 . . . 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 3071  cop 4635   class class class wbr 5149  {copab 5211  cfv 6544  Basecbs 17144  ltcplt 18261  ccvr 38180
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-covers 38184
This theorem is referenced by:  cvrlt  38188  cvrnbtwn  38189  cvrval2  38192  cvrcon3b  38195  lautcvr  39011
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