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Theorem ss0b 3391
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
StepHypRef Expression
1 0ss 3390 . . 3  |-  (/)  C_  A
2 eqss 3115 . . 3  |-  ( A  =  (/)  <->  ( A  C_  (/) 
/\  (/)  C_  A )
)
31, 2mpbiran2 890 . 2  |-  ( A  =  (/)  <->  A  C_  (/) )
43bicomi 195 1  |-  ( A 
C_  (/)  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1619    C_ wss 3078   (/)c0 3362
This theorem is referenced by:  ss0  3392  un00  3397  ssdisj  3411  pw0  3662  fnsuppeq0  5585  cnfcom2lem  7288  card0  7475  kmlem5  7664  cf0  7761  fin1a2lem12  7921  efgval  14861  ppttop  16576  0nnei  16681  sssu  24307  filnetlem4  25496  pnonsingN  28923  osumcllem4N  28949
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-dif 3081  df-in 3085  df-ss 3089  df-nul 3363
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