Theorem ne0i 3475

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2793. (Contributed by NM, 31-Dec-1993.)

Assertion

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

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3474	. 2 |- ( B e. A -> -. A = (/) )
2	1	neneqad 2457	1 |- ( B e. A -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2178   =/= wne 2378   (/)c0 3468

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-v 2778  df-dif 3176  df-nul 3469

This theorem is referenced by:  ne0d  3476  ne0ii  3478  vn0  3479  inelcm  3529  rzal  3566  rexn0  3567  snnzg  3760  prnz  3766  tpnz  3769  brne0  4109  onn0  4465  nn0eln0  4686  ordge1n0im  6545  nnmord  6626  map0g  6798  phpm  6988  fiintim  7054  addclpi  7475  mulclpi  7476  uzn0  9699  iccsupr  10123  pfxn0  11179  ringn0  13937