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Theorem shle0 28608
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 28606 . . 3 (𝐴S → 0𝐴)
21biantrud 529 . 2 (𝐴S → (𝐴 ⊆ 0 ↔ (𝐴 ⊆ 0 ∧ 0𝐴)))
3 eqss 3757 . 2 (𝐴 = 0 ↔ (𝐴 ⊆ 0 ∧ 0𝐴))
42, 3syl6bbr 278 1 (𝐴S → (𝐴 ⊆ 0𝐴 = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1630  wcel 2137  wss 3713   S csh 28092  0c0h 28099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-hilex 28163
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-br 4803  df-opab 4863  df-xp 5270  df-cnv 5272  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-sh 28371  df-ch0 28417
This theorem is referenced by:  chle0  28609  shne0i  28614  shs00i  28616  cdj3lem1  29600
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