Theorem ne0i 3453

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2776. (Contributed by NM, 31-Dec-1993.)

Assertion

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

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3452	. 2 |- ( B e. A -> -. A = (/) )
2	1	neneqad 2443	1 |- ( B e. A -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2164   =/= wne 2364   (/)c0 3446

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-v 2762  df-dif 3155  df-nul 3447

This theorem is referenced by:  ne0d  3454  ne0ii  3456  vn0  3457  inelcm  3507  rzal  3544  rexn0  3545  snnzg  3735  prnz  3740  tpnz  3743  brne0  4078  onn0  4429  nn0eln0  4648  ordge1n0im  6480  nnmord  6561  map0g  6733  phpm  6912  fiintim  6976  addclpi  7377  mulclpi  7378  uzn0  9598  iccsupr  10022  ringn0  13540