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Mirrors > Home > MPE Home > Th. List > elinel1 | Structured version Visualization version GIF version |
Description: Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
elinel1 | ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3796 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | |
2 | 1 | simplbi 476 | 1 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ∩ cin 3573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 |
This theorem is referenced by: elin1d 3802 inss1 3833 predel 5697 prmreclem2 15621 txkgen 21455 ncvsge0 22953 dchrisum0re 25202 uhgrspansubgrlem 26182 disjin 29399 xrge0tsmsd 29785 eulerpartgbij 30434 fiinfi 37878 gneispace 38432 elpwinss 39216 restuni3 39301 nel1nelin 39337 disjinfi 39380 inmap 39401 iocopn 39746 icoopn 39751 icomnfinre 39779 uzinico 39787 islpcn 39871 lptre2pt 39872 limcresiooub 39874 limcresioolb 39875 limsupmnflem 39952 limsupresxr 39998 liminfresxr 39999 liminfvalxr 40015 liminf0 40025 icccncfext 40100 stoweidlem39 40256 stoweidlem50 40267 stoweidlem57 40274 fourierdlem32 40356 fourierdlem33 40357 fourierdlem48 40371 fourierdlem49 40372 fourierdlem71 40394 sge0rnre 40581 sge00 40593 sge0tsms 40597 sge0cl 40598 sge0fsum 40604 sge0sup 40608 sge0less 40609 sge0gerp 40612 sge0resplit 40623 sge0split 40626 sge0iunmptlemre 40632 caragendifcl 40728 hoiqssbllem3 40838 hspmbllem2 40841 pimiooltgt 40921 pimdecfgtioc 40925 pimincfltioc 40926 pimdecfgtioo 40927 pimincfltioo 40928 sssmf 40947 smfaddlem1 40971 smfaddlem2 40972 smfadd 40973 mbfpsssmf 40991 smfmul 41002 smfdiv 41004 smfsuplem1 41017 smfliminflem 41036 fmtno4prm 41487 rngcid 41979 ringcid 42025 rhmsubclem3 42088 rhmsubcALTVlem3 42106 |
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