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Theorem cvrnbtwn 29530
Description: There is no element between the two arguments of the covers relation. (cvnbtwn 22980 analog.) (Contributed by NM, 18-Oct-2011.)
Hypotheses
Ref Expression
cvrfval.b  |-  B  =  ( Base `  K
)
cvrfval.s  |-  .<  =  ( lt `  K )
cvrfval.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
cvrnbtwn  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  X C Y )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) )

Proof of Theorem cvrnbtwn
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cvrfval.b . . . . 5  |-  B  =  ( Base `  K
)
2 cvrfval.s . . . . 5  |-  .<  =  ( lt `  K )
3 cvrfval.c . . . . 5  |-  C  =  (  <o  `  K )
41, 2, 3cvrval 29528 . . . 4  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( X  .<  Y  /\  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) ) ) )
543adant3r3 1162 . . 3  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X C Y  <->  ( X  .<  Y  /\  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) ) ) )
6 ralnex 2629 . . . . . . 7  |-  ( A. z  e.  B  -.  ( X  .<  z  /\  z  .<  Y )  <->  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) )
7 breq2 4108 . . . . . . . . . 10  |-  ( z  =  Z  ->  ( X  .<  z  <->  X  .<  Z ) )
8 breq1 4107 . . . . . . . . . 10  |-  ( z  =  Z  ->  (
z  .<  Y  <->  Z  .<  Y ) )
97, 8anbi12d 691 . . . . . . . . 9  |-  ( z  =  Z  ->  (
( X  .<  z  /\  z  .<  Y )  <-> 
( X  .<  Z  /\  Z  .<  Y ) ) )
109notbid 285 . . . . . . . 8  |-  ( z  =  Z  ->  ( -.  ( X  .<  z  /\  z  .<  Y )  <->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
1110rspcv 2956 . . . . . . 7  |-  ( Z  e.  B  ->  ( A. z  e.  B  -.  ( X  .<  z  /\  z  .<  Y )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
126, 11syl5bir 209 . . . . . 6  |-  ( Z  e.  B  ->  ( -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
1312adantld 453 . . . . 5  |-  ( Z  e.  B  ->  (
( X  .<  Y  /\  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
14133ad2ant3 978 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( ( X  .<  Y  /\  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
1514adantl 452 . . 3  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<  Y  /\  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
165, 15sylbid 206 . 2  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X C Y  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
17163impia 1148 1  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  X C Y )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   A.wral 2619   E.wrex 2620   class class class wbr 4104   ` cfv 5337   Basecbs 13245   ltcplt 14174    <o ccvr 29521
This theorem is referenced by:  cvrnbtwn2  29534  cvrnbtwn3  29535  cvrnbtwn4  29538  ltltncvr  29681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-covers 29525
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