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

Theorem lcvpss 39025
Description: The covers relation implies proper subset. (cvpss 32304 analog.) (Contributed by NM, 7-Jan-2015.)
Hypotheses
Ref Expression
lcvfbr.s 𝑆 = (LSubSp‘𝑊)
lcvfbr.c 𝐶 = ( ⋖L𝑊)
lcvfbr.w (𝜑𝑊𝑋)
lcvfbr.t (𝜑𝑇𝑆)
lcvfbr.u (𝜑𝑈𝑆)
lcvpss.d (𝜑𝑇𝐶𝑈)
Assertion
Ref Expression
lcvpss (𝜑𝑇𝑈)

Proof of Theorem lcvpss
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 lcvpss.d . . 3 (𝜑𝑇𝐶𝑈)
2 lcvfbr.s . . . 4 𝑆 = (LSubSp‘𝑊)
3 lcvfbr.c . . . 4 𝐶 = ( ⋖L𝑊)
4 lcvfbr.w . . . 4 (𝜑𝑊𝑋)
5 lcvfbr.t . . . 4 (𝜑𝑇𝑆)
6 lcvfbr.u . . . 4 (𝜑𝑈𝑆)
72, 3, 4, 5, 6lcvbr 39022 . . 3 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
81, 7mpbid 232 . 2 (𝜑 → (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
98simpld 494 1 (𝜑𝑇𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wrex 3070  wpss 3952   class class class wbr 5143  cfv 6561  LSubSpclss 20929  L clcv 39019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-lcv 39020
This theorem is referenced by:  lcvntr  39027  lcvat  39031  lsatcveq0  39033  lsat0cv  39034  lcvexchlem4  39038  lcvexchlem5  39039  lcv1  39042  lsatexch  39044  lsatcvat2  39052  islshpcv  39054
  Copyright terms: Public domain W3C validator