Theorem srhmsubclem1 20592

Description: Lemma 1 for srhmsubc 20595. (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 2819	. . . 4 ⊢ (𝑟 = 𝑋 → (𝑟 ∈ Ring ↔ 𝑋 ∈ Ring))
2		srhmsubc.s	. . . 4 ⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring
3	1, 2	vtoclri 3540	. . 3 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ Ring)
4	3	anim2i 617	. 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝑆) → (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ Ring))
5		srhmsubc.c	. . 3 ⊢ 𝐶 = (𝑈 ∩ 𝑆)
6	5	elin2 4150	. 2 ⊢ (𝑋 ∈ 𝐶 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝑆))
7		elin 3913	. 2 ⊢ (𝑋 ∈ (𝑈 ∩ Ring) ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ Ring))
8	4, 6, 7	3imtr4i 292	1 ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝑈 ∩ Ring))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ∩ cin 3896  Ringcrg 20151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-v 3438  df-in 3904

This theorem is referenced by:  srhmsubclem2  20593  srhmsubcALTVlem1  48433