Theorem sh0let 5365

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 5122	. . 3 |- (A e. SH -> 0v e. A)
2		snssi 1851	. . 3 |- (0v e. A -> {0v} (_ A)
3	1, 2	syl 12	. 2 |- (A e. SH -> {0v} (_ A)
4		df-ch0 5157	. 2 |- 0H = {0v}
5	3, 4	syl5ss 1544	1 |- (A e. SH -> 0H (_ A)

Syntax hints:   -> wi 2   e. wcel 1092   (_ wss 1487  {csn 1808  0vc0v 4961  SHcsh 4967  0Hc0h 4974

This theorem is referenced by:  ch0let 5366  shle0t 5367  orthin 5371  span0 5448

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-hilex 4983

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-sn 1811  df-sh 5114  df-ch0 5157

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