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

Theorem shle0 27478
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 27476 . . 3 (𝐴S → 0𝐴)
21biantrud 526 . 2 (𝐴S → (𝐴 ⊆ 0 ↔ (𝐴 ⊆ 0 ∧ 0𝐴)))
3 eqss 3582 . 2 (𝐴 = 0 ↔ (𝐴 ⊆ 0 ∧ 0𝐴))
42, 3syl6bbr 276 1 (𝐴S → (𝐴 ⊆ 0𝐴 = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wss 3539   S csh 26962  0c0h 26969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-hilex 27033
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5033  df-cnv 5035  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-sh 27241  df-ch0 27287
This theorem is referenced by:  chle0  27479  shne0i  27484  shs00i  27486  cdj3lem1  28470
  Copyright terms: Public domain W3C validator