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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 3967 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | |
| 2 | ne0i 4341 | . 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 2108 ≠ wne 2940 ∩ cin 3950 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-in 3958 df-nul 4334 |
| This theorem is referenced by: minel 4466 disji 5128 disjiun 5131 onnseq 8384 uniinqs 8837 en3lplem1 9652 cplem1 9929 fpwwe2lem11 10681 limsupgre 15517 cat1lem 18141 lmcls 23310 conncn 23434 iunconnlem 23435 conncompclo 23443 2ndcsep 23467 lfinpfin 23532 locfincmp 23534 txcls 23612 pthaus 23646 qtopeu 23724 trfbas2 23851 filss 23861 zfbas 23904 fmfnfm 23966 tsmsfbas 24136 restmetu 24583 qdensere 24790 reperflem 24840 reconnlem1 24848 metds0 24872 metnrmlem1a 24880 minveclem3b 25462 ovolicc2lem5 25556 taylfval 26400 wlk1walk 29657 wwlksm1edg 29901 disjif 32591 disjif2 32594 dfufd2lem 33577 subfacp1lem6 35190 erdszelem5 35200 pconnconn 35236 cvmseu 35281 neibastop2lem 36361 topdifinffinlem 37348 sstotbnd3 37783 brtrclfv2 43740 corcltrcl 43752 disjinfi 45197 |
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