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Theorem atssbase 35100
Description: The set of atoms is a subset of the base set. (atssch 29543 analog.) (Contributed by NM, 21-Oct-2011.)
Hypotheses
Ref Expression
atombase.b 𝐵 = (Base‘𝐾)
atombase.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
atssbase 𝐴𝐵

Proof of Theorem atssbase
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 atombase.b . . 3 𝐵 = (Base‘𝐾)
2 atombase.a . . 3 𝐴 = (Atoms‘𝐾)
31, 2atbase 35099 . 2 (𝑥𝐴𝑥𝐵)
43ssriv 3757 1 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wss 3724  cfv 6032  Basecbs 16065  Atomscatm 35073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5995  df-fun 6034  df-fv 6040  df-ats 35077
This theorem is referenced by:  atlatmstc  35129  atlatle  35130  pmapssbaN  35569  pmaple  35570  polsubN  35716  2polvalN  35723  2polssN  35724  3polN  35725  2pmaplubN  35735  paddunN  35736  poldmj1N  35737  pnonsingN  35742  ispsubcl2N  35756  psubclinN  35757  paddatclN  35758  polsubclN  35761  poml4N  35762
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