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Theorem cvrval 33370
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 28331 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 33369 . . . . 5 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))})
5 3anass 1034 . . . . . 6 (((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)) ↔ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))))
65opabbii 4643 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}
74, 6syl6eq 2659 . . . 4 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))})
87breqd 4588 . . 3 (𝐾𝐴 → (𝑋𝐶𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌))
983ad2ant1 1074 . 2 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌))
10 df-br 4578 . . . 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 4580 . . . . . 6 (𝑥 = 𝑋 → (𝑥 < 𝑦𝑋 < 𝑦))
12 breq1 4580 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 < 𝑧𝑋 < 𝑧))
1312anbi1d 736 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑥 < 𝑧𝑧 < 𝑦) ↔ (𝑋 < 𝑧𝑧 < 𝑦)))
1413rexbidv 3033 . . . . . . 7 (𝑥 = 𝑋 → (∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦) ↔ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦)))
1514notbid 306 . . . . . 6 (𝑥 = 𝑋 → (¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦) ↔ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦)))
1611, 15anbi12d 742 . . . . 5 (𝑥 = 𝑋 → ((𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)) ↔ (𝑋 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦))))
17 breq2 4581 . . . . . 6 (𝑦 = 𝑌 → (𝑋 < 𝑦𝑋 < 𝑌))
18 breq2 4581 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑧 < 𝑦𝑧 < 𝑌))
1918anbi2d 735 . . . . . . . 8 (𝑦 = 𝑌 → ((𝑋 < 𝑧𝑧 < 𝑦) ↔ (𝑋 < 𝑧𝑧 < 𝑌)))
2019rexbidv 3033 . . . . . . 7 (𝑦 = 𝑌 → (∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦) ↔ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)))
2120notbid 306 . . . . . 6 (𝑦 = 𝑌 → (¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦) ↔ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)))
2217, 21anbi12d 742 . . . . 5 (𝑦 = 𝑌 → ((𝑋 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑦)) ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
2316, 22opelopab2 4911 . . . 4 ((𝑋𝐵𝑌𝐵) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))} ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
2410, 23syl5bb 270 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
25243adant1 1071 . 2 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
269, 25bitrd 266 1 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wrex 2896  cop 4130   class class class wbr 4577  {copab 4636  cfv 5790  Basecbs 15641  ltcplt 16710  ccvr 33363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-covers 33367
This theorem is referenced by:  cvrlt  33371  cvrnbtwn  33372  cvrval2  33375  cvrcon3b  33378  lautcvr  34192
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