Theorem ss0b 3326

Description: Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)

Assertion

Ref	Expression
ss0b	|- ( A C_ (/) <-> A = (/) )

Proof of Theorem ss0b

Step	Hyp	Ref	Expression
1		0ss 3325	. . 3 |- (/) C_ A
2		eqss 3041	. . 3 |- ( A = (/) <-> ( A C_ (/) /\ (/) C_ A ) )
3	1, 2	mpbiran2 888	. 2 |- ( A = (/) <-> A C_ (/) )
4	3	bicomi 131	1 |- ( A C_ (/) <-> A = (/) )

Syntax hints:   <-> wb 104   = wceq 1290   C_ wss 3000   (/)c0 3287

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-dif 3002  df-in 3006  df-ss 3013  df-nul 3288

This theorem is referenced by:  ss0  3327  un00  3333  ssdisj  3343  pw0  3590  card0  6877