Theorem ne0i 3457

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 3456	. 2 |- ( B e. A -> -. A = (/) )
2	1	neneqad 2446	1 |- ( B e. A -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2167   =/= wne 2367   (/)c0 3450

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 3451

This theorem is referenced by:  ne0d  3458  ne0ii  3460  vn0  3461  inelcm  3511  rzal  3548  rexn0  3549  snnzg  3739  prnz  3744  tpnz  3747  brne0  4082  onn0  4435  nn0eln0  4656  ordge1n0im  6494  nnmord  6575  map0g  6747  phpm  6926  fiintim  6990  addclpi  7392  mulclpi  7393  uzn0  9614  iccsupr  10038  ringn0  13592