Theorem n0i 3514

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2830. (Contributed by NM, 31-Dec-1993.)

Assertion

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

Proof of Theorem n0i

Step	Hyp	Ref	Expression
1		noel 3512	. . 3 |- -. B e. (/)
2		eleq2 2296	. . 3 |- ( A = (/) -> ( B e. A <-> B e. (/) ) )
3	1, 2	mtbiri 682	. 2 |- ( A = (/) -> -. B e. A )
4	3	con2i 632	1 |- ( B e. A -> -. A = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1398   e. wcel 2203   (/)c0 3508

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-dif 3213  df-nul 3509

This theorem is referenced by:  ne0i  3515  n0ii  3517  unidif0  4280  iin0r  4282  nnm00  6763  dif1enen  7137  enq0tr  7749  gsum0g  13609  gsumval2  13610