Theorem sh0let 9279

Description: The zero subspace is the smallest subspace.

Assertion

Ref	Expression
sh0let	|- (A e. SH -> 0H (_ A)

Proof of Theorem sh0let

Step	Hyp	Ref	Expression
1		sh0 9005	. . 3 |- (A e. SH -> 0h e. A)
2		snssi 2457	. . 3 |- (0h e. A -> {0h} (_ A)
3	1, 2	syl 10	. 2 |- (A e. SH -> {0h} (_ A)
4		df-ch0 9046	. 2 |- 0H = {0h}
5	3, 4	syl5ss 2095	1 |- (A e. SH -> 0H (_ A)

Syntax hints:   -> wi 3   e. wcel 955   (_ wss 2037  {csn 2399  0hc0v 8730  SHcsh 8736  0Hc0h 8743

This theorem is referenced by:  ch0let 9280  shle0t 9281  orthin 9285  shs0 9287  span0 9380

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-hilex 8790

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-v 1803  df-in 2041  df-ss 2043  df-sn 2402  df-sh 8997  df-ch0 9046

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