Theorem ne0i 3429

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2753. (Contributed by NM, 31-Dec-1993.)

Assertion

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

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3428	. 2 |- ( B e. A -> -. A = (/) )
2	1	neneqad 2426	1 |- ( B e. A -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2148   =/= wne 2347   (/)c0 3422

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 2739  df-dif 3131  df-nul 3423

This theorem is referenced by:  ne0d  3430  ne0ii  3432  vn0  3433  inelcm  3483  rzal  3520  rexn0  3521  snnzg  3709  prnz  3714  tpnz  3717  brne0  4052  onn0  4400  nn0eln0  4619  ordge1n0im  6436  nnmord  6517  map0g  6687  phpm  6864  fiintim  6927  addclpi  7325  mulclpi  7326  uzn0  9542  iccsupr  9965  ringn0  13235