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Theorem atssbase 37041
Description: The set of atoms is a subset of the base set. (atssch 30424 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 37040 . 2 (𝑥𝐴𝑥𝐵)
43ssriv 3905 1 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  wss 3866  cfv 6380  Basecbs 16760  Atomscatm 37014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-ats 37018
This theorem is referenced by:  atlatmstc  37070  atlatle  37071  pmapssbaN  37511  pmaple  37512  polsubN  37658  2polvalN  37665  2polssN  37666  3polN  37667  2pmaplubN  37677  paddunN  37678  poldmj1N  37679  pnonsingN  37684  ispsubcl2N  37698  psubclinN  37699  paddatclN  37700  polsubclN  37703  poml4N  37704
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