Theorem inelcm 3498

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 3333	. 2 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
2		ne0i 3444	. 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 2160   =/= wne 2360   i^i cin 3143   (/)c0 3437

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-v 2754  df-dif 3146  df-in 3150  df-nul 3438

This theorem is referenced by:  minel  3499  disjiun  4013