Theorem inelcm 3552

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 3387	. 2 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
2		ne0i 3498	. 2 |- ( A e. ( B i^i C ) -> ( B i^i C ) =/= (/) )
3	1, 2	sylbir 135	1 |- ( ( A e. B /\ A e. C ) -> ( B i^i C ) =/= (/) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   =/= wne 2400   i^i cin 3196   (/)c0 3491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2801  df-dif 3199  df-in 3203  df-nul 3492

This theorem is referenced by:  minel  3553  disjiun  4078