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Theorem srhmsubclem1 20694
Description: Lemma 1 for srhmsubc 20697. (Contributed by AV, 19-Feb-2020.)
Hypotheses
Ref Expression
srhmsubc.s 𝑟𝑆 𝑟 ∈ Ring
srhmsubc.c 𝐶 = (𝑈𝑆)
Assertion
Ref Expression
srhmsubclem1 (𝑋𝐶𝑋 ∈ (𝑈 ∩ Ring))
Distinct variable groups:   𝑆,𝑟   𝑋,𝑟
Allowed substitution hints:   𝐶(𝑟)   𝑈(𝑟)

Proof of Theorem srhmsubclem1
StepHypRef Expression
1 eleq1 2827 . . . 4 (𝑟 = 𝑋 → (𝑟 ∈ Ring ↔ 𝑋 ∈ Ring))
2 srhmsubc.s . . . 4 𝑟𝑆 𝑟 ∈ Ring
31, 2vtoclri 3590 . . 3 (𝑋𝑆𝑋 ∈ Ring)
43anim2i 617 . 2 ((𝑋𝑈𝑋𝑆) → (𝑋𝑈𝑋 ∈ Ring))
5 srhmsubc.c . . 3 𝐶 = (𝑈𝑆)
65elin2 4213 . 2 (𝑋𝐶 ↔ (𝑋𝑈𝑋𝑆))
7 elin 3979 . 2 (𝑋 ∈ (𝑈 ∩ Ring) ↔ (𝑋𝑈𝑋 ∈ Ring))
84, 6, 73imtr4i 292 1 (𝑋𝐶𝑋 ∈ (𝑈 ∩ Ring))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  cin 3962  Ringcrg 20251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-in 3970
This theorem is referenced by:  srhmsubclem2  20695  srhmsubcALTVlem1  48167
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