Theorem shle0t 9361

Description: No subspace is smaller than the zero subspace.

Assertion

Ref	Expression
shle0t	|- (A e. SH -> (A (_ 0H <-> A = 0H))

Proof of Theorem shle0t

Step	Hyp	Ref	Expression
1		sh0let 9359	. . 3 |- (A e. SH -> 0H (_ A)
2	1	biantrud 728	. 2 |- (A e. SH -> (A (_ 0H <-> (A (_ 0H /\ 0H (_ A)))
3		eqss 2080	. 2 |- (A = 0H <-> (A (_ 0H /\ 0H (_ A))
4	2, 3	syl6bbr 540	1 |- (A e. SH -> (A (_ 0H <-> A = 0H))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960   (_ wss 2050  SHcsh 8792  0Hc0h 8799

This theorem is referenced by:  chle0t 9362  shne0 9366  shs00 9368  cdj3lem1 10356

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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