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Mirrors > Home > MPE Home > Th. List > Mathboxes > srhmsubclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for srhmsubc 42076. (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 2689 | . . . 4 ⊢ (𝑟 = 𝑋 → (𝑟 ∈ Ring ↔ 𝑋 ∈ Ring)) | |
2 | srhmsubc.s | . . . 4 ⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring | |
3 | 1, 2 | vtoclri 3283 | . . 3 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ Ring) |
4 | 3 | anim2i 593 | . 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝑆) → (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ Ring)) |
5 | srhmsubc.c | . . 3 ⊢ 𝐶 = (𝑈 ∩ 𝑆) | |
6 | 5 | elin2 3801 | . 2 ⊢ (𝑋 ∈ 𝐶 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝑆)) |
7 | elin 3796 | . 2 ⊢ (𝑋 ∈ (𝑈 ∩ Ring) ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ Ring)) | |
8 | 4, 6, 7 | 3imtr4i 281 | 1 ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝑈 ∩ Ring)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∩ cin 3573 Ringcrg 18547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-v 3202 df-in 3581 |
This theorem is referenced by: srhmsubclem2 42074 srhmsubcALTVlem1 42092 |
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