Theorem ne0i 3467

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2788. (Contributed by NM, 31-Dec-1993.)

Assertion

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

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3466	. 2 |- ( B e. A -> -. A = (/) )
2	1	neneqad 2455	1 |- ( B e. A -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2176   =/= wne 2376   (/)c0 3460

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-v 2774  df-dif 3168  df-nul 3461

This theorem is referenced by:  ne0d  3468  ne0ii  3470  vn0  3471  inelcm  3521  rzal  3558  rexn0  3559  snnzg  3750  prnz  3755  tpnz  3758  brne0  4093  onn0  4447  nn0eln0  4668  ordge1n0im  6522  nnmord  6603  map0g  6775  phpm  6962  fiintim  7028  addclpi  7440  mulclpi  7441  uzn0  9664  iccsupr  10088  ringn0  13822