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Theorem cvrlt 29460
Description: The covers relation implies the less-than relation. (cvpss 22865 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 29459 . 2  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( X  .<  Y  /\  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) ) ) )
54simprbda 606 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255   Basecbs 13148   ltcplt 14075    <o ccvr 29452
This theorem is referenced by:  ncvr1  29462  cvrletrN  29463  cvrnbtwn2  29465  cvrnbtwn3  29466  cvrle  29468  cvrnle  29470  cvrne  29471  0ltat  29481  atlen0  29500  atcvreq0  29504  cvlcvr1  29529  cvrval3  29602  cvrval4N  29603  cvrexchlem  29608  ltcvrntr  29613  cvrntr  29614  cvrat2  29618  atltcvr  29624  1cvratex  29662  ps-2  29667  llnnleat  29702  lplnnle2at  29730  lvolnle3at  29771  lhp0lt  30192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-covers 29456
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