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Mirrors > Home > ILE Home > Th. List > ssrin | GIF version |
Description: Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
ssrin | ⊢ (A ⊆ B → (A ∩ 𝐶) ⊆ (B ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 2933 | . . . 4 ⊢ (A ⊆ B → (x ∈ A → x ∈ B)) | |
2 | 1 | anim1d 319 | . . 3 ⊢ (A ⊆ B → ((x ∈ A ∧ x ∈ 𝐶) → (x ∈ B ∧ x ∈ 𝐶))) |
3 | elin 3120 | . . 3 ⊢ (x ∈ (A ∩ 𝐶) ↔ (x ∈ A ∧ x ∈ 𝐶)) | |
4 | elin 3120 | . . 3 ⊢ (x ∈ (B ∩ 𝐶) ↔ (x ∈ B ∧ x ∈ 𝐶)) | |
5 | 2, 3, 4 | 3imtr4g 194 | . 2 ⊢ (A ⊆ B → (x ∈ (A ∩ 𝐶) → x ∈ (B ∩ 𝐶))) |
6 | 5 | ssrdv 2945 | 1 ⊢ (A ⊆ B → (A ∩ 𝐶) ⊆ (B ∩ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 ∩ cin 2910 ⊆ wss 2911 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-in 2918 df-ss 2925 |
This theorem is referenced by: sslin 3157 ss2in 3158 ssdisj 3271 ssdifin0 3298 ssres 4580 |
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