Theorem srhmsubclem1 44351

Description: Lemma 1 for srhmsubc 44354. (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 2902	. . . 4 ⊢ (𝑟 = 𝑋 → (𝑟 ∈ Ring ↔ 𝑋 ∈ Ring))
2		srhmsubc.s	. . . 4 ⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring
3	1, 2	vtoclri 3587	. . 3 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ Ring)
4	3	anim2i 618	. 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝑆) → (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ Ring))
5		srhmsubc.c	. . 3 ⊢ 𝐶 = (𝑈 ∩ 𝑆)
6	5	elin2 4176	. 2 ⊢ (𝑋 ∈ 𝐶 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝑆))
7		elin 4171	. 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 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140   ∩ cin 3937  Ringcrg 19299

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-v 3498  df-in 3945

This theorem is referenced by:  srhmsubclem2  44352  srhmsubcALTVlem1  44370