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Theorem shle0 30682
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 30680 . . 3 (𝐴S → 0𝐴)
21biantrud 532 . 2 (𝐴S → (𝐴 ⊆ 0 ↔ (𝐴 ⊆ 0 ∧ 0𝐴)))
3 eqss 3996 . 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 396   = wceq 1541  wcel 2106  wss 3947   S csh 30168  0c0h 30175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-hilex 30239
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-sh 30447  df-ch0 30493
This theorem is referenced by:  chle0  30683  shne0i  30688  shs00i  30690  cdj3lem1  31674
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