Theorem ss0b 3500

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 3499	. . 3 |- (/) C_ A
2		eqss 3208	. . 3 |- ( A = (/) <-> ( A C_ (/) /\ (/) C_ A ) )
3	1, 2	mpbiran2 944	. 2 |- ( A = (/) <-> A C_ (/) )
4	3	bicomi 132	1 |- ( A C_ (/) <-> A = (/) )

Syntax hints:   <-> wb 105   = wceq 1373   C_ wss 3166   (/)c0 3460

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3461

This theorem is referenced by:  ss0  3501  un00  3507  ssdisj  3517  pw0  3780  card0  7295  0nnei  14625