Theorem ne0i 3431

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2755. (Contributed by NM, 31-Dec-1993.)

Assertion

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

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3430	. 2 |- ( B e. A -> -. A = (/) )
2	1	neneqad 2426	1 |- ( B e. A -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2148   =/= wne 2347   (/)c0 3424

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 2741  df-dif 3133  df-nul 3425

This theorem is referenced by:  ne0d  3432  ne0ii  3434  vn0  3435  inelcm  3485  rzal  3522  rexn0  3523  snnzg  3711  prnz  3716  tpnz  3719  brne0  4054  onn0  4402  nn0eln0  4621  ordge1n0im  6440  nnmord  6521  map0g  6691  phpm  6868  fiintim  6931  addclpi  7329  mulclpi  7330  uzn0  9546  iccsupr  9969  ringn0  13243