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Theorem cvrlt 29757
Description: The covers relation implies the less-than relation. (cvpss 23745 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 29756 . 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 2671   class class class wbr 4176   ` cfv 5417   Basecbs 13428   ltcplt 14357    <o ccvr 29749
This theorem is referenced by:  ncvr1  29759  cvrletrN  29760  cvrnbtwn2  29762  cvrnbtwn3  29763  cvrle  29765  cvrnle  29767  cvrne  29768  0ltat  29778  atlen0  29797  atcvreq0  29801  cvlcvr1  29826  cvrval3  29899  cvrval4N  29900  cvrexchlem  29905  ltcvrntr  29910  cvrntr  29911  cvrat2  29915  atltcvr  29921  1cvratex  29959  ps-2  29964  llnnleat  29999  lplnnle2at  30027  lvolnle3at  30068  lhp0lt  30489
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 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-covers 29753
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