Theorem srhmsubclem1 20729

Description: Lemma 1 for srhmsubc 20732. (Contributed by AV, 19-Feb-2020.)

Hypotheses

Ref	Expression
srhmsubc.s	⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring
srhmsubc.c	⊢ 𝐶 = (𝑈 ∩ 𝑆)

Assertion

Ref	Expression
srhmsubclem1	⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝑈 ∩ Ring))

Distinct variable groups:   𝑆,𝑟   𝑋,𝑟

Allowed substitution hints:   𝐶(𝑟)   𝑈(𝑟)

Proof of Theorem srhmsubclem1

Step	Hyp	Ref	Expression
1		eleq1 2852	. . . 4 ⊢ (𝑟 = 𝑋 → (𝑟 ∈ Ring ↔ 𝑋 ∈ Ring))
2		srhmsubc.s	. . . 4 ⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring
3	1, 2	vtoclri 3551	. . 3 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ Ring)
4	3	anim2i 626	. 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝑆) → (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ Ring))
5		srhmsubc.c	. . 3 ⊢ 𝐶 = (𝑈 ∩ 𝑆)
6	5	elin2 4157	. 2 ⊢ (𝑋 ∈ 𝐶 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝑆))
7		elin 3922	. 2 ⊢ (𝑋 ∈ (𝑈 ∩ Ring) ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ Ring))
8	4, 6, 7	3imtr4i 294	1 ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝑈 ∩ Ring))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078   ∩ cin 3905  Ringcrg 20285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-v 3458  df-in 3913

This theorem is referenced by:  srhmsubclem2  20730  srhmsubcALTVlem1  48950