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Theorem linindsv 42028
 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
StepHypRef Expression
1 rellininds 42026 . . 3 Rel linIndS
21brrelexi 5082 . 2 (𝑆 linIndS 𝑀𝑆 ∈ V)
31brrelex2i 5083 . 2 (𝑆 linIndS 𝑀𝑀 ∈ V)
42, 3jca 553 1 (𝑆 linIndS 𝑀 → (𝑆 ∈ V ∧ 𝑀 ∈ V))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583   linIndS clininds 42023 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-lininds 42025 This theorem is referenced by:  linindsi  42030
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