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Theorem ch0lei 31479
Description: The closed subspace zero is the smallest member of C. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
Hypothesis
Ref Expression
ch0le.1 𝐴C
Assertion
Ref Expression
ch0lei 0𝐴

Proof of Theorem ch0lei
StepHypRef Expression
1 ch0le.1 . 2 𝐴C
2 ch0le 31469 . 2 (𝐴C → 0𝐴)
31, 2ax-mp 5 1 0𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  wss 3962   C cch 30957  0c0h 30963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-hilex 31027
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-xp 5694  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fv 6570  df-ov 7433  df-sh 31235  df-ch 31249  df-ch0 31281
This theorem is referenced by:  chj0i  31483  chm0i  31518  hst0  32261
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