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Theorem lcvpss 33329
Description: The covers relation implies proper subset. (cvpss 28528 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 33326 . . 3 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
81, 7mpbid 221 . 2 (𝜑 → (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
98simpld 474 1 (𝜑𝑇𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wcel 1977  wrex 2897  wpss 3541   class class class wbr 4583  cfv 5804  LSubSpclss 18753  L clcv 33323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-lcv 33324
This theorem is referenced by:  lcvntr  33331  lcvat  33335  lsatcveq0  33337  lsat0cv  33338  lcvexchlem4  33342  lcvexchlem5  33343  lcv1  33346  lsatexch  33348  lsatcvat2  33356  islshpcv  33358
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