Theorem ne0i 3430

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2754. (Contributed by NM, 31-Dec-1993.)

Assertion

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

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3429	. 2 |- ( B e. A -> -. A = (/) )
2	1	neneqad 2426	1 |- ( B e. A -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2148   =/= wne 2347   (/)c0 3423

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-v 2740  df-dif 3132  df-nul 3424

This theorem is referenced by:  ne0d  3431  ne0ii  3433  vn0  3434  inelcm  3484  rzal  3521  rexn0  3522  snnzg  3710  prnz  3715  tpnz  3718  brne0  4053  onn0  4401  nn0eln0  4620  ordge1n0im  6437  nnmord  6518  map0g  6688  phpm  6865  fiintim  6928  addclpi  7326  mulclpi  7327  uzn0  9543  iccsupr  9966  ringn0  13237