Theorem ne0i 3458

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2779. (Contributed by NM, 31-Dec-1993.)

Assertion

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

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3457	. 2 |- ( B e. A -> -. A = (/) )
2	1	neneqad 2446	1 |- ( B e. A -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2167   =/= wne 2367   (/)c0 3451

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-v 2765  df-dif 3159  df-nul 3452

This theorem is referenced by:  ne0d  3459  ne0ii  3461  vn0  3462  inelcm  3512  rzal  3549  rexn0  3550  snnzg  3740  prnz  3745  tpnz  3748  brne0  4083  onn0  4436  nn0eln0  4657  ordge1n0im  6495  nnmord  6576  map0g  6748  phpm  6927  fiintim  6993  addclpi  7396  mulclpi  7397  uzn0  9619  iccsupr  10043  ringn0  13626