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Theorem atssbase 39292
Description: The set of atoms is a subset of the base set. (atssch 32363 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 39291 . 2 (𝑥𝐴𝑥𝐵)
43ssriv 3986 1 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wss 3950  cfv 6560  Basecbs 17248  Atomscatm 39265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ats 39269
This theorem is referenced by:  atlatmstc  39321  atlatle  39322  pmapssbaN  39763  pmaple  39764  polsubN  39910  2polvalN  39917  2polssN  39918  3polN  39919  2pmaplubN  39929  paddunN  39930  poldmj1N  39931  pnonsingN  39936  ispsubcl2N  39950  psubclinN  39951  paddatclN  39952  polsubclN  39955  poml4N  39956
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