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Theorem cvrlt 29512
Description: The covers relation implies the less-than relation. (cvpss 22973 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 29511 . 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 1642    e. wcel 1710   E.wrex 2620   class class class wbr 4102   ` cfv 5334   Basecbs 13239   ltcplt 14168    <o ccvr 29504
This theorem is referenced by:  ncvr1  29514  cvrletrN  29515  cvrnbtwn2  29517  cvrnbtwn3  29518  cvrle  29520  cvrnle  29522  cvrne  29523  0ltat  29533  atlen0  29552  atcvreq0  29556  cvlcvr1  29581  cvrval3  29654  cvrval4N  29655  cvrexchlem  29660  ltcvrntr  29665  cvrntr  29666  cvrat2  29670  atltcvr  29676  1cvratex  29714  ps-2  29719  llnnleat  29754  lplnnle2at  29782  lvolnle3at  29823  lhp0lt  30244
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 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
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 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-iota 5298  df-fun 5336  df-fv 5342  df-covers 29508
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