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Theorem srhmsubclem1 44272
Description: Lemma 1 for srhmsubc 44275. (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 2897 . . . 4 (𝑟 = 𝑋 → (𝑟 ∈ Ring ↔ 𝑋 ∈ Ring))
2 srhmsubc.s . . . 4 𝑟𝑆 𝑟 ∈ Ring
31, 2vtoclri 3582 . . 3 (𝑋𝑆𝑋 ∈ Ring)
43anim2i 616 . 2 ((𝑋𝑈𝑋𝑆) → (𝑋𝑈𝑋 ∈ Ring))
5 srhmsubc.c . . 3 𝐶 = (𝑈𝑆)
65elin2 4171 . 2 (𝑋𝐶 ↔ (𝑋𝑈𝑋𝑆))
7 elin 4166 . 2 (𝑋 ∈ (𝑈 ∩ Ring) ↔ (𝑋𝑈𝑋 ∈ Ring))
84, 6, 73imtr4i 293 1 (𝑋𝐶𝑋 ∈ (𝑈 ∩ Ring))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  cin 3932  Ringcrg 19226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-v 3494  df-in 3940
This theorem is referenced by:  srhmsubclem2  44273  srhmsubcALTVlem1  44291
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