Theorem lindepsnlininds 49154

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 5118	. . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑠 linIndS 𝑚 ↔ 𝑆 linIndS 𝑀))
2	1	notbid 321	. 2 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (¬ 𝑠 linIndS 𝑚 ↔ ¬ 𝑆 linIndS 𝑀))
3		df-lindeps 49146	. 2 ⊢ linDepS = {⟨𝑠, 𝑚⟩ ∣ ¬ 𝑠 linIndS 𝑚}
4	2, 3	brabga 5521	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   class class class wbr 5113   linIndS clininds 49142   linDepS clindeps 49143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5407

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-lindeps 49146

This theorem is referenced by:  islindeps  49155  islininds2  49186