Theorem linindsv 46030

Description: The classes of the module and its linearly independent subsets are sets. (Contributed by AV, 13-Apr-2019.)

Assertion

Ref	Expression
linindsv	⊢ (𝑆 linIndS 𝑀 → (𝑆 ∈ V ∧ 𝑀 ∈ V))

Proof of Theorem linindsv

Step	Hyp	Ref	Expression
1		rellininds 46028	. 2 ⊢ Rel linIndS
2	1	brrelex12i 5653	1 ⊢ (𝑆 linIndS 𝑀 → (𝑆 ∈ V ∧ 𝑀 ∈ V))

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2104  Vcvv 3437   class class class wbr 5081   linIndS clininds 46025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-xp 5606  df-rel 5607  df-lininds 46027

This theorem is referenced by:  linindsi  46032