Theorem ne0i 3420

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2746. (Contributed by NM, 31-Dec-1993.)

Assertion

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

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3419	. 2 |- ( B e. A -> -. A = (/) )
2	1	neneqad 2419	1 |- ( B e. A -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2141   =/= wne 2340   (/)c0 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-nul 3415

This theorem is referenced by:  ne0d  3421  ne0ii  3423  vn0  3424  inelcm  3474  rzal  3511  rexn0  3512  snnzg  3698  prnz  3703  tpnz  3706  brne0  4036  onn0  4383  nn0eln0  4602  ordge1n0im  6412  nnmord  6493  map0g  6662  phpm  6839  fiintim  6902  addclpi  7276  mulclpi  7277  uzn0  9489  iccsupr  9910