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Theorem ss0b 3445
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 3444 . . 3  |-  (/)  C_  A
2 eqss 3155 . . 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 3113   (/)c0 3416
This theorem is referenced by:  ss0  3446  un00  3451  ssdisj  3465  pw0  3722  fnsuppeq0  5653  cnfcom2lem  7358  card0  7545  kmlem5  7734  cf0  7831  fin1a2lem12  7991  mreexexlem3d  13496  efgval  14974  ppttop  16692  0nnei  16797  sssu  24494  filnetlem4  25683  pnonsingN  29273  osumcllem4N  29299
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 1927  ax-ext 2237
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 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2759  df-dif 3116  df-in 3120  df-ss 3127  df-nul 3417
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