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Theorem lcvpss 39006
Description: The covers relation implies proper subset. (cvpss 32314 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 39003 . . 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 1537  wcel 2106  wrex 3068  wpss 3964   class class class wbr 5148  cfv 6563  LSubSpclss 20947  L clcv 39000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-lcv 39001
This theorem is referenced by:  lcvntr  39008  lcvat  39012  lsatcveq0  39014  lsat0cv  39015  lcvexchlem4  39019  lcvexchlem5  39020  lcv1  39023  lsatexch  39025  lsatcvat2  39033  islshpcv  39035
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