Theorem lindepsnlininds 48928

Description: A linearly dependent subset is not a linearly independent subset. (Contributed by AV, 26-Apr-2019.)

Assertion

Ref	Expression
lindepsnlininds	⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))

Proof of Theorem lindepsnlininds

Dummy variables 𝑚 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq12 5090	. . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑠 linIndS 𝑚 ↔ 𝑆 linIndS 𝑀))
2	1	notbid 318	. 2 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (¬ 𝑠 linIndS 𝑚 ↔ ¬ 𝑆 linIndS 𝑀))
3		df-lindeps 48920	. 2 ⊢ linDepS = {⟨𝑠, 𝑚⟩ ∣ ¬ 𝑠 linIndS 𝑚}
4	2, 3	brabga 5489	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   class class class wbr 5085   linIndS clininds 48916   linDepS clindeps 48917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-lindeps 48920

This theorem is referenced by:  islindeps  48929  islininds2  48960