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Theorem ch0lei 22955
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  |-  A  e. 
CH
Assertion
Ref Expression
ch0lei  |-  0H  C_  A

Proof of Theorem ch0lei
StepHypRef Expression
1 ch0le.1 . 2  |-  A  e. 
CH
2 ch0le 22945 . 2  |-  ( A  e.  CH  ->  0H  C_  A )
31, 2ax-mp 8 1  |-  0H  C_  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1726    C_ wss 3322   CHcch 22434   0Hc0h 22440
This theorem is referenced by:  chj0i  22959  chm0i  22994  hst0  23738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-hilex 22504
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-cnv 4888  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fv 5464  df-ov 6086  df-sh 22711  df-ch 22726  df-ch0 22757
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