Theorem linindsv 47383

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 47381	. 2 ⊢ Rel linIndS
2	1	brrelex12i 5724	1 ⊢ (𝑆 linIndS 𝑀 → (𝑆 ∈ V ∧ 𝑀 ∈ V))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2098  Vcvv 3468   class class class wbr 5141   linIndS clininds 47378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-lininds 47380

This theorem is referenced by:  linindsi  47385