Theorem n0i 3414

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2742. (Contributed by NM, 31-Dec-1993.)

Assertion

Ref	Expression
n0i	|- ( B e. A -> -. A = (/) )

Proof of Theorem n0i

Step	Hyp	Ref	Expression
1		noel 3413	. . 3 |- -. B e. (/)
2		eleq2 2230	. . 3 |- ( A = (/) -> ( B e. A <-> B e. (/) ) )
3	1, 2	mtbiri 665	. 2 |- ( A = (/) -> -. B e. A )
4	3	con2i 617	1 |- ( B e. A -> -. A = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1343   e. wcel 2136   (/)c0 3409

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-nul 3410

This theorem is referenced by:  ne0i  3415  n0ii  3417  unidif0  4146  iin0r  4148  nnm00  6497  dif1enen  6846  enq0tr  7375