Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcvpss Structured version   Unicode version

Theorem lcvpss 29822
Description: The covers relation implies proper subset. (cvpss 23788 analog.) (Contributed by NM, 7-Jan-2015.)
Hypotheses
Ref Expression
lcvfbr.s  |-  S  =  ( LSubSp `  W )
lcvfbr.c  |-  C  =  (  <oLL  `  W )
lcvfbr.w  |-  ( ph  ->  W  e.  X )
lcvfbr.t  |-  ( ph  ->  T  e.  S )
lcvfbr.u  |-  ( ph  ->  U  e.  S )
lcvpss.d  |-  ( ph  ->  T C U )
Assertion
Ref Expression
lcvpss  |-  ( ph  ->  T  C.  U )

Proof of Theorem lcvpss
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 lcvpss.d . . 3  |-  ( ph  ->  T C U )
2 lcvfbr.s . . . 4  |-  S  =  ( LSubSp `  W )
3 lcvfbr.c . . . 4  |-  C  =  (  <oLL  `  W )
4 lcvfbr.w . . . 4  |-  ( ph  ->  W  e.  X )
5 lcvfbr.t . . . 4  |-  ( ph  ->  T  e.  S )
6 lcvfbr.u . . . 4  |-  ( ph  ->  U  e.  S )
72, 3, 4, 5, 6lcvbr 29819 . . 3  |-  ( ph  ->  ( T C U  <-> 
( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U
) ) ) )
81, 7mpbid 202 . 2  |-  ( ph  ->  ( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U
) ) )
98simpld 446 1  |-  ( ph  ->  T  C.  U )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706    C. wpss 3321   class class class wbr 4212   ` cfv 5454   LSubSpclss 16008    <oLL clcv 29816
This theorem is referenced by:  lcvntr  29824  lcvat  29828  lsatcveq0  29830  lsat0cv  29831  lcvexchlem4  29835  lcvexchlem5  29836  lcv1  29839  lsatexch  29841  lsatcvat2  29849  islshpcv  29851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-lcv 29817
  Copyright terms: Public domain W3C validator