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Mirrors > Home > ILE Home > Th. List > inss1 | GIF version |
Description: The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.) |
Ref | Expression |
---|---|
inss1 | ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3229 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | 1 | simplbi 272 | . 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐴) |
3 | 2 | ssriv 3071 | 1 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 ∩ cin 3040 ⊆ wss 3041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-in 3047 df-ss 3054 |
This theorem is referenced by: inss2 3267 ssinss1 3275 unabs 3277 inssddif 3287 inv1 3369 disjdif 3405 inundifss 3410 relin1 4627 resss 4813 resmpt3 4838 cnvcnvss 4963 funin 5164 funimass2 5171 fnresin1 5207 fnres 5209 fresin 5271 ssimaex 5450 fneqeql2 5497 isoini2 5688 ofrfval 5958 ofvalg 5959 ofrval 5960 off 5962 ofres 5964 ofco 5968 smores 6157 smores2 6159 tfrlem5 6179 pmresg 6538 unfiin 6782 sbthlem7 6819 peano5nnnn 7668 peano5nni 8687 rexanuz 10715 fvsetsid 11904 tgvalex 12130 tgval2 12131 eltg3 12137 tgcl 12144 tgdom 12152 tgidm 12154 epttop 12170 ntropn 12197 ntrin 12204 cnptopresti 12318 cnptoprest 12319 txcnmpt 12353 xmetres 12462 metres 12463 blin2 12512 metrest 12586 tgioo 12626 limcresi 12715 |
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