Theorem linindsv 48728

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 48726	. 2 ⊢ Rel linIndS
2	1	brrelex12i 5678	1 ⊢ (𝑆 linIndS 𝑀 → (𝑆 ∈ V ∧ 𝑀 ∈ V))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Vcvv 3439   class class class wbr 5097   linIndS clininds 48723

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 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

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 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5629  df-rel 5630  df-lininds 48725

This theorem is referenced by:  linindsi  48730