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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 3758 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | |
2 | ne0i 3880 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅) | |
3 | 1, 2 | sylbir 224 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ≠ wne 2780 ∩ cin 3539 ∅c0 3874 |
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-ne 2782 df-v 3175 df-dif 3543 df-in 3547 df-nul 3875 |
This theorem is referenced by: minel 3985 minelOLD 3986 disji 4570 disjiun 4573 onnseq 7328 uniinqs 7714 en3lplem1 8394 cplem1 8635 fpwwe2lem12 9342 limsupgre 14060 lmcls 20916 concn 21039 iunconlem 21040 concompclo 21048 2ndcsep 21072 lfinpfin 21137 locfincmp 21139 txcls 21217 pthaus 21251 qtopeu 21329 trfbas2 21457 filss 21467 zfbas 21510 fmfnfm 21572 tsmsfbas 21741 restmetu 22185 qdensere 22383 reperflem 22429 reconnlem1 22437 metds0 22461 metnrmlem1a 22469 minveclem3b 23007 ovolicc2lem5 23096 taylfval 23917 wwlkm1edg 26263 disjif 28773 disjif2 28776 subfacp1lem6 30421 erdszelem5 30431 pconcon 30467 cvmseu 30512 neibastop2lem 31525 topdifinffinlem 32371 sstotbnd3 32745 brtrclfv2 37038 corcltrcl 37050 disjinfi 38375 1wlk1walk 40843 wwlksm1edg 41078 |
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