Theorem ss0b 3486

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 3485	. . 3 |- (/) C_ A
2		eqss 3194	. . 3 |- ( A = (/) <-> ( A C_ (/) /\ (/) C_ A ) )
3	1, 2	mpbiran2 943	. 2 |- ( A = (/) <-> A C_ (/) )
4	3	bicomi 132	1 |- ( A C_ (/) <-> A = (/) )

Syntax hints:   <-> wb 105   = wceq 1364   C_ wss 3153   (/)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-v 2762  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3447

This theorem is referenced by:  ss0  3487  un00  3493  ssdisj  3503  pw0  3765  card0  7238  0nnei  14298