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Mirrors > Home > MPE Home > Th. List > ssinss1 | Structured version Visualization version GIF version |
Description: Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.) |
Ref | Expression |
---|---|
ssinss1 | ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 3795 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | sstr2 3575 | . 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3539 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 |
This theorem is referenced by: inss 3804 wfrlem4 7305 wfrlem5 7306 fipwuni 8215 ssfin4 9015 distop 20610 fctop 20618 cctop 20620 ntrin 20675 innei 20739 lly1stc 21109 txcnp 21233 isfild 21472 utoptop 21848 restmetu 22185 lecmi 27845 mdslj2i 28563 mdslmd1lem1 28568 mdslmd1lem2 28569 elpwincl1 28741 pnfneige0 29325 inelcarsg 29700 ballotlemfrc 29915 bnj1177 30328 bnj1311 30346 frrlem4 31027 frrlem5 31028 cldbnd 31491 neiin 31497 ontgval 31600 mblfinlem4 32619 pmodlem1 34150 pmodlem2 34151 pmod1i 34152 pmod2iN 34153 pmodl42N 34155 dochdmj1 35697 ssficl 36893 ntrclskb 37387 ntrclsk13 37389 ntrneik3 37414 ntrneik13 37416 ssinss1d 38239 icccncfext 38773 fourierdlem48 39047 fourierdlem49 39048 fourierdlem113 39112 caragendifcl 39404 omelesplit 39408 carageniuncllem2 39412 carageniuncl 39413 |
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