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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 3300 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | 1 | simplbi 272 | . 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐴) |
3 | 2 | ssriv 3141 | 1 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2135 ∩ cin 3110 ⊆ wss 3111 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-in 3117 df-ss 3124 |
This theorem is referenced by: inss2 3338 ssinss1 3346 unabs 3348 inssddif 3358 inv1 3440 disjdif 3476 inundifss 3481 relin1 4716 resss 4902 resmpt3 4927 cnvcnvss 5052 funin 5253 funimass2 5260 fnresin1 5296 fnres 5298 fresin 5360 ssimaex 5541 fneqeql2 5588 isoini2 5781 ofrfval 6052 ofvalg 6053 ofrval 6054 off 6056 ofres 6058 ofco 6062 smores 6251 smores2 6253 tfrlem5 6273 pmresg 6633 unfiin 6882 sbthlem7 6919 peano5nnnn 7824 peano5nni 8851 rexanuz 10916 nninfdclemcl 12326 nninfdclemp1 12328 fvsetsid 12371 tgvalex 12597 tgval2 12598 eltg3 12604 tgcl 12611 tgdom 12619 tgidm 12621 epttop 12637 ntropn 12664 ntrin 12671 cnptopresti 12785 cnptoprest 12786 txcnmpt 12820 xmetres 12929 metres 12930 blin2 12979 metrest 13053 tgioo 13093 limcresi 13182 bj-charfun 13530 bj-charfundc 13531 bj-charfundcALT 13532 |
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