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

Theorem shle0 29555
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 29553 . . 3 (𝐴S → 0𝐴)
21biantrud 535 . 2 (𝐴S → (𝐴 ⊆ 0 ↔ (𝐴 ⊆ 0 ∧ 0𝐴)))
3 eqss 3933 . 2 (𝐴 = 0 ↔ (𝐴 ⊆ 0 ∧ 0𝐴))
42, 3bitr4di 292 1 (𝐴S → (𝐴 ⊆ 0𝐴 = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2112  wss 3883   S csh 29041  0c0h 29048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710  ax-sep 5209  ax-hilex 29112
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-rab 3073  df-v 3425  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4255  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5575  df-cnv 5577  df-dm 5579  df-rn 5580  df-res 5581  df-ima 5582  df-sh 29320  df-ch0 29366
This theorem is referenced by:  chle0  29556  shne0i  29561  shs00i  29563  cdj3lem1  30547
  Copyright terms: Public domain W3C validator