Theorem inelcm 3418

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

Assertion

Ref	Expression
inelcm	|- ( ( A e. B /\ A e. C ) -> ( B i^i C ) =/= (/) )

Proof of Theorem inelcm

Step	Hyp	Ref	Expression
1		elin 3254	. 2 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
2		ne0i 3364	. 2 |- ( A e. ( B i^i C ) -> ( B i^i C ) =/= (/) )
3	1, 2	sylbir 134	1 |- ( ( A e. B /\ A e. C ) -> ( B i^i C ) =/= (/) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1480   =/= wne 2306   i^i cin 3065   (/)c0 3358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-v 2683  df-dif 3068  df-in 3072  df-nul 3359

This theorem is referenced by:  minel  3419  disjiun  3919