Theorem sh0let 9359

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 9079	. . 3 |- (A e. SH -> 0h e. A)
2		snssi 2470	. . 3 |- (0h e. A -> {0h} (_ A)
3	1, 2	syl 10	. 2 |- (A e. SH -> {0h} (_ A)
4		df-ch0 9120	. 2 |- 0H = {0h}
5	3, 4	syl5ss 2108	1 |- (A e. SH -> 0H (_ A)

Syntax hints:   -> wi 3   e. wcel 960   (_ wss 2050  {csn 2413  0hc0v 8786  SHcsh 8792  0Hc0h 8799

This theorem is referenced by:  ch0let 9360  shle0t 9361  orthin 9365  shs0 9367  span0 9460

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-hilex 8864

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-in 2054  df-ss 2056  df-sn 2416  df-sh 9071  df-ch0 9120

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