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Theorem ch0lei 29241
 Description: The closed subspace zero is the smallest member of Cℋ. (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 29231 . 2 (𝐴C → 0𝐴)
31, 2ax-mp 5 1 0𝐴
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2111   ⊆ wss 3881   Cℋ cch 28719  0ℋc0h 28725 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-hilex 28789 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fv 6332  df-ov 7138  df-sh 28997  df-ch 29011  df-ch0 29043 This theorem is referenced by:  chj0i  29245  chm0i  29280  hst0  30023
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