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Theorem atssbase 36293
Description: The set of atoms is a subset of the base set. (atssch 30034 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 36292 . 2 (𝑥𝐴𝑥𝐵)
43ssriv 3974 1 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wss 3939  cfv 6351  Basecbs 16475  Atomscatm 36266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-iota 6311  df-fun 6353  df-fv 6359  df-ats 36270
This theorem is referenced by:  atlatmstc  36322  atlatle  36323  pmapssbaN  36763  pmaple  36764  polsubN  36910  2polvalN  36917  2polssN  36918  3polN  36919  2pmaplubN  36929  paddunN  36930  poldmj1N  36931  pnonsingN  36936  ispsubcl2N  36950  psubclinN  36951  paddatclN  36952  polsubclN  36955  poml4N  36956
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