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Theorem lindepsnlininds 48219
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 5154 . . 3 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑠 linIndS 𝑚𝑆 linIndS 𝑀))
21notbid 318 . 2 ((𝑠 = 𝑆𝑚 = 𝑀) → (¬ 𝑠 linIndS 𝑚 ↔ ¬ 𝑆 linIndS 𝑀))
3 df-lindeps 48211 . 2 linDepS = {⟨𝑠, 𝑚⟩ ∣ ¬ 𝑠 linIndS 𝑚}
42, 3brabga 5536 1 ((𝑆𝑉𝑀𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1535  wcel 2104   class class class wbr 5149   linIndS clininds 48207   linDepS clindeps 48208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-lindeps 48211
This theorem is referenced by:  islindeps  48220  islininds2  48251
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