Theorem inelcm 4372

Description: The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)

Assertion

Ref	Expression
inelcm	⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅)

Proof of Theorem inelcm

Step	Hyp	Ref	Expression
1		elin 3897	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
2		ne0i 4250	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅)
3	1, 2	sylbir 238	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111   ≠ wne 2987   ∩ cin 3880  ∅c0 4243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-v 3443  df-dif 3884  df-in 3888  df-nul 4244

This theorem is referenced by:  minel  4373  disji  5013  disjiun  5017  onnseq  7964  uniinqs  8360  en3lplem1  9059  cplem1  9302  fpwwe2lem12  10052  limsupgre  14830  lmcls  21907  conncn  22031  iunconnlem  22032  conncompclo  22040  2ndcsep  22064  lfinpfin  22129  locfincmp  22131  txcls  22209  pthaus  22243  qtopeu  22321  trfbas2  22448  filss  22458  zfbas  22501  fmfnfm  22563  tsmsfbas  22733  restmetu  23177  qdensere  23375  reperflem  23423  reconnlem1  23431  metds0  23455  metnrmlem1a  23463  minveclem3b  24032  ovolicc2lem5  24125  taylfval  24954  wlk1walk  27428  wwlksm1edg  27667  disjif  30341  disjif2  30344  subfacp1lem6  32545  erdszelem5  32555  pconnconn  32591  cvmseu  32636  neibastop2lem  33821  topdifinffinlem  34764  sstotbnd3  35214  brtrclfv2  40428  corcltrcl  40440  disjinfi  41820