Theorem atssbase 39409

Description: The set of atoms is a subset of the base set. (atssch 32325 analog.) (Contributed by NM, 21-Oct-2011.)

Hypotheses

Ref	Expression
atombase.b	⊢ 𝐵 = (Base‘𝐾)
atombase.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
atssbase	⊢ 𝐴 ⊆ 𝐵

Proof of Theorem atssbase

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		atombase.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		atombase.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
3	1, 2	atbase 39408	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)
4	3	ssriv 3934	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   = wceq 1541   ⊆ wss 3898  ‘cfv 6486  Basecbs 17122  Atomscatm 39382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fv 6494  df-ats 39386

This theorem is referenced by:  atlatmstc  39438  atlatle  39439  pmapssbaN  39879  pmaple  39880  polsubN  40026  2polvalN  40033  2polssN  40034  3polN  40035  2pmaplubN  40045  paddunN  40046  poldmj1N  40047  pnonsingN  40052  ispsubcl2N  40066  psubclinN  40067  paddatclN  40068  polsubclN  40071  poml4N  40072