Theorem lindepsnlininds 42241

Description: A linearly dependent subset is not a linearly independent subset. (Contributed by AV, 26-Apr-2019.)

Assertion

Ref	Expression
lindepsnlininds	|- ( ( S e. V /\ M e. 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 4658	. . 3 |- ( ( s = S /\ m = M ) -> ( s linIndS m <-> S linIndS M ) )
2	1	notbid 308	. 2 |- ( ( s = S /\ m = M ) -> ( -. s linIndS m <-> -. S linIndS M ) )
3		df-lindeps 42233	. 2 |- linDepS = { <. s , m >. | -. s linIndS m }
4	2, 3	brabga 4989	1 |- ( ( S e. V /\ M e. W ) -> ( S linDepS M <-> -. S linIndS M ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   linIndS clininds 42229   linDepS clindeps 42230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-lindeps 42233

This theorem is referenced by:  islindeps  42242  islininds2  42273