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Theorem lcvpss 36040
Description: The covers relation implies proper subset. (cvpss 29989 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 36037 . . 3 (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
81, 7mpbid 233 . 2 (𝜑 → (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈)))
98simpld 495 1 (𝜑𝑇𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wrex 3136  wpss 3934   class class class wbr 5057  cfv 6348  LSubSpclss 19632  L clcv 36034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-lcv 36035
This theorem is referenced by:  lcvntr  36042  lcvat  36046  lsatcveq0  36048  lsat0cv  36049  lcvexchlem4  36053  lcvexchlem5  36054  lcv1  36057  lsatexch  36059  lsatcvat2  36067  islshpcv  36069
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