HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  shle0 Structured version   Visualization version   GIF version

Theorem shle0 30960
Description: No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
Assertion
Ref Expression
shle0 (𝐴S → (𝐴 ⊆ 0𝐴 = 0))

Proof of Theorem shle0
StepHypRef Expression
1 sh0le 30958 . . 3 (𝐴S → 0𝐴)
21biantrud 530 . 2 (𝐴S → (𝐴 ⊆ 0 ↔ (𝐴 ⊆ 0 ∧ 0𝐴)))
3 eqss 3998 . 2 (𝐴 = 0 ↔ (𝐴 ⊆ 0 ∧ 0𝐴))
42, 3bitr4di 288 1 (𝐴S → (𝐴 ⊆ 0𝐴 = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  wss 3949   S csh 30446  0c0h 30453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300  ax-hilex 30517
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-sh 30725  df-ch0 30771
This theorem is referenced by:  chle0  30961  shne0i  30966  shs00i  30968  cdj3lem1  31952
  Copyright terms: Public domain W3C validator