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Theorem lindepsnlininds 43256
Description: A linearly dependent subset is not a linearly independent subset. (Contributed by AV, 26-Apr-2019.)
Assertion
Ref Expression
lindepsnlininds ((𝑆𝑉𝑀𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))

Proof of Theorem lindepsnlininds
Dummy variables 𝑚 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq12 4891 . . 3 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑠 linIndS 𝑚𝑆 linIndS 𝑀))
21notbid 310 . 2 ((𝑠 = 𝑆𝑚 = 𝑀) → (¬ 𝑠 linIndS 𝑚 ↔ ¬ 𝑆 linIndS 𝑀))
3 df-lindeps 43248 . 2 linDepS = {⟨𝑠, 𝑚⟩ ∣ ¬ 𝑠 linIndS 𝑚}
42, 3brabga 5226 1 ((𝑆𝑉𝑀𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107   class class class wbr 4886   linIndS clininds 43244   linDepS clindeps 43245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-lindeps 43248
This theorem is referenced by:  islindeps  43257  islininds2  43288
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