Theorem srhmsubclem1 20831

Description: Lemma 1 for srhmsubc 20834. (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 2853	. . . 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 629	. 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝑆) → (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ Ring))
5		srhmsubc.c	. . 3 ⊢ 𝐶 = (𝑈 ∩ 𝑆)
6	5	elin2 4156	. 2 ⊢ (𝑋 ∈ 𝐶 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝑆))
7		elin 3922	. 2 ⊢ (𝑋 ∈ (𝑈 ∩ Ring) ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ Ring))
8	4, 6, 7	3imtr4i 295	1 ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝑈 ∩ Ring))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ∩ cin 3905  Ringcrg 20364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-v 3459  df-in 3913

This theorem is used by:  srhmsubclem2  20832  srhmsubcALTVlem1  49165