Theorem shle0 28301

Description: No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
shle0	|- ( A e. SH -> ( A C_ 0H <-> A = 0H ) )

Proof of Theorem shle0

Step	Hyp	Ref	Expression
1		sh0le 28299	. . 3 |- ( A e. SH -> 0H C_ A )
2	1	biantrud 528	. 2 |- ( A e. SH -> ( A C_ 0H <-> ( A C_ 0H /\ 0H C_ A ) ) )
3		eqss 3618	. 2 |- ( A = 0H <-> ( A C_ 0H /\ 0H C_ A ) )
4	2, 3	syl6bbr 278	1 |- ( A e. SH -> ( A C_ 0H <-> A = 0H ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   SHcsh 27785   0Hc0h 27792

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-hilex 27856

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-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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-sh 28064  df-ch0 28110

This theorem is referenced by:  chle0  28302  shne0i  28307  shs00i  28309  cdj3lem1  29293