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| Description: Lemma 1 for srhmsubc 20681. (Contributed by AV, 19-Feb-2020.) | 
| Ref | Expression | 
|---|---|
| srhmsubc.s | ⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring | 
| srhmsubc.c | ⊢ 𝐶 = (𝑈 ∩ 𝑆) | 
| Ref | Expression | 
|---|---|
| srhmsubclem1 | ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝑈 ∩ Ring)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eleq1 2828 | . . . 4 ⊢ (𝑟 = 𝑋 → (𝑟 ∈ Ring ↔ 𝑋 ∈ Ring)) | |
| 2 | srhmsubc.s | . . . 4 ⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring | |
| 3 | 1, 2 | vtoclri 3589 | . . 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 4202 | . 2 ⊢ (𝑋 ∈ 𝐶 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝑆)) | 
| 7 | elin 3966 | . 2 ⊢ (𝑋 ∈ (𝑈 ∩ Ring) ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ Ring)) | |
| 8 | 4, 6, 7 | 3imtr4i 292 | 1 ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝑈 ∩ Ring)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∩ cin 3949 Ringcrg 20231 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-v 3481 df-in 3957 | 
| This theorem is referenced by: srhmsubclem2 20679 srhmsubcALTVlem1 48244 | 
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