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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 3264 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | 1 | simplbi 272 | . 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐴) |
3 | 2 | ssriv 3106 | 1 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 ∩ cin 3075 ⊆ wss 3076 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-in 3082 df-ss 3089 |
This theorem is referenced by: inss2 3302 ssinss1 3310 unabs 3312 inssddif 3322 inv1 3404 disjdif 3440 inundifss 3445 relin1 4665 resss 4851 resmpt3 4876 cnvcnvss 5001 funin 5202 funimass2 5209 fnresin1 5245 fnres 5247 fresin 5309 ssimaex 5490 fneqeql2 5537 isoini2 5728 ofrfval 5998 ofvalg 5999 ofrval 6000 off 6002 ofres 6004 ofco 6008 smores 6197 smores2 6199 tfrlem5 6219 pmresg 6578 unfiin 6822 sbthlem7 6859 peano5nnnn 7724 peano5nni 8747 rexanuz 10792 fvsetsid 12032 tgvalex 12258 tgval2 12259 eltg3 12265 tgcl 12272 tgdom 12280 tgidm 12282 epttop 12298 ntropn 12325 ntrin 12332 cnptopresti 12446 cnptoprest 12447 txcnmpt 12481 xmetres 12590 metres 12591 blin2 12640 metrest 12714 tgioo 12754 limcresi 12843 |
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