Theorem inelcm 3362

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 3198	. 2 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
2		ne0i 3308	. 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 1445   =/= wne 2262   i^i cin 3012   (/)c0 3302

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-v 2635  df-dif 3015  df-in 3019  df-nul 3303

This theorem is referenced by:  minel  3363  disjiun  3862