Theorem ne0i 3295

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2638. (Contributed by NM, 31-Dec-1993.)

Assertion

Ref	Expression
ne0i	|- ( B e. A -> A =/= (/) )

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3294	. 2 |- ( B e. A -> -. A = (/) )
2	1	neneqad 2335	1 |- ( B e. A -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 1439   =/= wne 2256   (/)c0 3289

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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-v 2624  df-dif 3004  df-nul 3290

This theorem is referenced by:  ne0d  3296  ne0ii  3298  vn0  3299  inelcm  3349  rzal  3385  rexn0  3386  snnzg  3565  prnz  3570  tpnz  3573  onn0  4238  nn0eln0  4448  ordge1n0im  6216  nnmord  6292  map0g  6461  phpm  6637  fiintim  6695  addclpi  6949  mulclpi  6950  uzn0  9097  iccsupr  9447