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Mirrors > Home > MPE Home > Th. List > Mathboxes > srhmsubclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for srhmsubc 46927. (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 2821 | . . . 4 ⊢ (𝑟 = 𝑋 → (𝑟 ∈ Ring ↔ 𝑋 ∈ Ring)) | |
2 | srhmsubc.s | . . . 4 ⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring | |
3 | 1, 2 | vtoclri 3576 | . . 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 4196 | . 2 ⊢ (𝑋 ∈ 𝐶 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝑆)) |
7 | elin 3963 | . 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 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∩ cin 3946 Ringcrg 20049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-v 3476 df-in 3954 |
This theorem is referenced by: srhmsubclem2 46925 srhmsubcALTVlem1 46943 |
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