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Theorem cvrlt 29753
Description: The covers relation implies the less-than relation. (cvpss 23741 analog.) (Contributed by NM, 8-Oct-2011.)
Hypotheses
Ref Expression
cvrfval.b  |-  B  =  ( Base `  K
)
cvrfval.s  |-  .<  =  ( lt `  K )
cvrfval.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
cvrlt  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X C Y )  ->  X  .<  Y )

Proof of Theorem cvrlt
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cvrfval.b . . 3  |-  B  =  ( Base `  K
)
2 cvrfval.s . . 3  |-  .<  =  ( lt `  K )
3 cvrfval.c . . 3  |-  C  =  (  <o  `  K )
41, 2, 3cvrval 29752 . 2  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( X  .<  Y  /\  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) ) ) )
54simprbda 607 1  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X C Y )  ->  X  .<  Y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2667   class class class wbr 4172   ` cfv 5413   Basecbs 13424   ltcplt 14353    <o ccvr 29745
This theorem is referenced by:  ncvr1  29755  cvrletrN  29756  cvrnbtwn2  29758  cvrnbtwn3  29759  cvrle  29761  cvrnle  29763  cvrne  29764  0ltat  29774  atlen0  29793  atcvreq0  29797  cvlcvr1  29822  cvrval3  29895  cvrval4N  29896  cvrexchlem  29901  ltcvrntr  29906  cvrntr  29907  cvrat2  29911  atltcvr  29917  1cvratex  29955  ps-2  29960  llnnleat  29995  lplnnle2at  30023  lvolnle3at  30064  lhp0lt  30485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-covers 29749
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