Theorem lindepsnlininds 45681

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 5075	. . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑠 linIndS 𝑚 ↔ 𝑆 linIndS 𝑀))
2	1	notbid 317	. 2 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (¬ 𝑠 linIndS 𝑚 ↔ ¬ 𝑆 linIndS 𝑀))
3		df-lindeps 45673	. 2 ⊢ linDepS = {⟨𝑠, 𝑚⟩ ∣ ¬ 𝑠 linIndS 𝑚}
4	2, 3	brabga 5440	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   class class class wbr 5070   linIndS clininds 45669   linDepS clindeps 45670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-lindeps 45673

This theorem is referenced by:  islindeps  45682  islininds2  45713