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Mirrors > Home > MPE Home > Th. List > inelcm | Structured version Visualization version GIF version |
Description: The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.) |
Ref | Expression |
---|---|
inelcm | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3979 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | |
2 | ne0i 4347 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅) | |
3 | 1, 2 | sylbir 235 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ≠ wne 2938 ∩ cin 3962 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-v 3480 df-dif 3966 df-in 3970 df-nul 4340 |
This theorem is referenced by: minel 4472 disji 5133 disjiun 5136 onnseq 8383 uniinqs 8836 en3lplem1 9650 cplem1 9927 fpwwe2lem11 10679 limsupgre 15514 cat1lem 18150 lmcls 23326 conncn 23450 iunconnlem 23451 conncompclo 23459 2ndcsep 23483 lfinpfin 23548 locfincmp 23550 txcls 23628 pthaus 23662 qtopeu 23740 trfbas2 23867 filss 23877 zfbas 23920 fmfnfm 23982 tsmsfbas 24152 restmetu 24599 qdensere 24806 reperflem 24854 reconnlem1 24862 metds0 24886 metnrmlem1a 24894 minveclem3b 25476 ovolicc2lem5 25570 taylfval 26415 wlk1walk 29672 wwlksm1edg 29911 disjif 32598 disjif2 32601 dfufd2lem 33557 subfacp1lem6 35170 erdszelem5 35180 pconnconn 35216 cvmseu 35261 neibastop2lem 36343 topdifinffinlem 37330 sstotbnd3 37763 brtrclfv2 43717 corcltrcl 43729 disjinfi 45135 |
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