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Mirrors > Home > MPE Home > Th. List > ssrind | Structured version Visualization version GIF version |
Description: Add right intersection to subclass relation. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
ssrind.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
ssrind | ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrind.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | ssrin 4249 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3961 ⊆ wss 3962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-in 3969 df-ss 3979 |
This theorem is referenced by: fictb 10281 isacs1i 17701 rescabs 17882 rescabsOLD 17883 lsmdisj 19713 dmdprdsplit2lem 20079 rhmsscrnghm 20681 rngcresringcat 20685 acsfn1p 20816 obselocv 21765 restbas 23181 neitr 23203 restcls 23204 restntr 23205 nrmsep 23380 cldllycmp 23518 fclsneii 24040 tsmsres 24167 trcfilu 24318 metdseq0 24889 iundisj2 25597 uniioombllem3 25633 ppisval 27161 ppisval2 27162 chtwordi 27213 ppiwordi 27219 chpub 27278 chebbnd1lem1 27527 mdbr2 32324 mdslj1i 32347 mdsl2i 32350 mdslmd1lem1 32353 mdslmd3i 32360 mdexchi 32363 sumdmdlem 32446 iundisj2f 32609 iundisj2fi 32804 cycpmco2f1 33126 tocyccntz 33146 esumrnmpt2 34048 bnj1177 34998 sstotbnd2 37760 lcvexchlem5 39019 pnonsingN 39915 dochnoncon 41373 eldioph2lem2 42748 limsupres 45660 limsupresxr 45721 liminfresxr 45722 liminflelimsuplem 45730 ssdisjd 48655 |
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