Theorem n0i 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
n0i	|- ( B e. A -> -. A = (/) )

Proof of Theorem n0i

Step	Hyp	Ref	Expression
1		noel 3418	. . 3 |- -. B e. (/)
2		eleq2 2234	. . 3 |- ( A = (/) -> ( B e. A <-> B e. (/) ) )
3	1, 2	mtbiri 670	. 2 |- ( A = (/) -> -. B e. A )
4	3	con2i 622	1 |- ( B e. A -> -. A = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1348   e. wcel 2141   (/)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-v 2732  df-dif 3123  df-nul 3415

This theorem is referenced by:  ne0i  3421  n0ii  3423  unidif0  4153  iin0r  4155  nnm00  6509  dif1enen  6858  enq0tr  7396