Theorem n0i 3456

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2779. (Contributed by NM, 31-Dec-1993.)

Assertion

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

Proof of Theorem n0i

Step	Hyp	Ref	Expression
1		noel 3454	. . 3 |- -. B e. (/)
2		eleq2 2260	. . 3 |- ( A = (/) -> ( B e. A <-> B e. (/) ) )
3	1, 2	mtbiri 676	. 2 |- ( A = (/) -> -. B e. A )
4	3	con2i 628	1 |- ( B e. A -> -. A = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   e. wcel 2167   (/)c0 3450

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-nul 3451

This theorem is referenced by:  ne0i  3457  n0ii  3459  unidif0  4200  iin0r  4202  nnm00  6588  dif1enen  6941  enq0tr  7501  gsum0g  13039  gsumval2  13040