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Theorem srhmsubclem1 20609
Description: Lemma 1 for srhmsubc 20612. (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 2813 . . . 4 (𝑟 = 𝑋 → (𝑟 ∈ Ring ↔ 𝑋 ∈ Ring))
2 srhmsubc.s . . . 4 𝑟𝑆 𝑟 ∈ Ring
31, 2vtoclri 3567 . . 3 (𝑋𝑆𝑋 ∈ Ring)
43anim2i 615 . 2 ((𝑋𝑈𝑋𝑆) → (𝑋𝑈𝑋 ∈ Ring))
5 srhmsubc.c . . 3 𝐶 = (𝑈𝑆)
65elin2 4192 . 2 (𝑋𝐶 ↔ (𝑋𝑈𝑋𝑆))
7 elin 3957 . 2 (𝑋 ∈ (𝑈 ∩ Ring) ↔ (𝑋𝑈𝑋 ∈ Ring))
84, 6, 73imtr4i 291 1 (𝑋𝐶𝑋 ∈ (𝑈 ∩ Ring))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3051  cin 3940  Ringcrg 20172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-v 3465  df-in 3948
This theorem is referenced by:  srhmsubclem2  20610  srhmsubcALTVlem1  47493
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