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Theorem subrngss 20444
Description: A subring is a subset. (Contributed by AV, 14-Feb-2025.)
Hypothesis
Ref Expression
subrngss.1 𝐵 = (Base‘𝑅)
Assertion
Ref Expression
subrngss (𝐴 ∈ (SubRng‘𝑅) → 𝐴𝐵)

Proof of Theorem subrngss
StepHypRef Expression
1 subrngss.1 . . 3 𝐵 = (Base‘𝑅)
21issubrng 20443 . 2 (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵))
32simp3bi 1146 1 (𝐴 ∈ (SubRng‘𝑅) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  wss 3948  cfv 6543  (class class class)co 7412  Basecbs 17151  s cress 17180  Rngcrng 20053  SubRngcsubrng 20441
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-subrng 20442
This theorem is referenced by:  subrngsubg  20448  subrngmre  20458  subsubrng  20459  rhmimasubrnglem  20461  rhmimasubrng  20462
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