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Theorem atssbase 39001
Description: The set of atoms is a subset of the base set. (atssch 32273 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 39000 . 2 (𝑥𝐴𝑥𝐵)
43ssriv 3982 1 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wss 3946  cfv 6546  Basecbs 17208  Atomscatm 38974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-iota 6498  df-fun 6548  df-fv 6554  df-ats 38978
This theorem is referenced by:  atlatmstc  39030  atlatle  39031  pmapssbaN  39472  pmaple  39473  polsubN  39619  2polvalN  39626  2polssN  39627  3polN  39628  2pmaplubN  39638  paddunN  39639  poldmj1N  39640  pnonsingN  39645  ispsubcl2N  39659  psubclinN  39660  paddatclN  39661  polsubclN  39664  poml4N  39665
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