Theorem ss0b 3508

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 3507	. . 3 |- (/) C_ A
2		eqss 3216	. . 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 3174   (/)c0 3468

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-nul 3469

This theorem is referenced by:  ss0  3509  un00  3515  ssdisj  3525  pw0  3791  card0  7321  0nnei  14740