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Mirrors > Home > MPE Home > Th. List > srhmsubclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for srhmsubc 20612. (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 2813 | . . . 4 ⊢ (𝑟 = 𝑋 → (𝑟 ∈ Ring ↔ 𝑋 ∈ Ring)) | |
2 | srhmsubc.s | . . . 4 ⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring | |
3 | 1, 2 | vtoclri 3567 | . . 3 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ Ring) |
4 | 3 | anim2i 615 | . 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝑆) → (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ Ring)) |
5 | srhmsubc.c | . . 3 ⊢ 𝐶 = (𝑈 ∩ 𝑆) | |
6 | 5 | elin2 4192 | . 2 ⊢ (𝑋 ∈ 𝐶 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝑆)) |
7 | elin 3957 | . 2 ⊢ (𝑋 ∈ (𝑈 ∩ Ring) ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ Ring)) | |
8 | 4, 6, 7 | 3imtr4i 291 | 1 ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝑈 ∩ Ring)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ∩ cin 3940 Ringcrg 20172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-v 3465 df-in 3948 |
This theorem is referenced by: srhmsubclem2 20610 srhmsubcALTVlem1 47493 |
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