Theorem inelcm 3557

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 3392	. 2 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
2		ne0i 3503	. 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 2202   =/= wne 2403   i^i cin 3200   (/)c0 3496

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-v 2805  df-dif 3203  df-in 3207  df-nul 3497

This theorem is referenced by:  minel  3558  disjiun  4088