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Theorem atssbase 38463
Description: The set of atoms is a subset of the base set. (atssch 31863 analog.) (Contributed by NM, 21-Oct-2011.)
Hypotheses
Ref Expression
atombase.b 𝐡 = (Baseβ€˜πΎ)
atombase.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
atssbase 𝐴 βŠ† 𝐡

Proof of Theorem atssbase
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 atombase.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 atombase.a . . 3 𝐴 = (Atomsβ€˜πΎ)
31, 2atbase 38462 . 2 (π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ 𝐡)
43ssriv 3985 1 𝐴 βŠ† 𝐡
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539   βŠ† wss 3947  β€˜cfv 6542  Basecbs 17148  Atomscatm 38436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ats 38440
This theorem is referenced by:  atlatmstc  38492  atlatle  38493  pmapssbaN  38934  pmaple  38935  polsubN  39081  2polvalN  39088  2polssN  39089  3polN  39090  2pmaplubN  39100  paddunN  39101  poldmj1N  39102  pnonsingN  39107  ispsubcl2N  39121  psubclinN  39122  paddatclN  39123  polsubclN  39126  poml4N  39127
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