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Theorem atssbase 39272
Description: The set of atoms is a subset of the base set. (atssch 32372 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 39271 . 2 (𝑥𝐴𝑥𝐵)
43ssriv 3999 1 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wss 3963  cfv 6563  Basecbs 17245  Atomscatm 39245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ats 39249
This theorem is referenced by:  atlatmstc  39301  atlatle  39302  pmapssbaN  39743  pmaple  39744  polsubN  39890  2polvalN  39897  2polssN  39898  3polN  39899  2pmaplubN  39909  paddunN  39910  poldmj1N  39911  pnonsingN  39916  ispsubcl2N  39930  psubclinN  39931  paddatclN  39932  polsubclN  39935  poml4N  39936
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