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Theorem ss0b 3649
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 3648 . . 3  |-  (/)  C_  A
2 eqss 3355 . . 3  |-  ( A  =  (/)  <->  ( A  C_  (/) 
/\  (/)  C_  A )
)
31, 2mpbiran2 886 . 2  |-  ( A  =  (/)  <->  A  C_  (/) )
43bicomi 194 1  |-  ( A 
C_  (/)  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    C_ wss 3312   (/)c0 3620
This theorem is referenced by:  ss0  3650  un00  3655  ssdisj  3669  pw0  3937  fnsuppeq0  5944  cnfcom2lem  7647  card0  7834  kmlem5  8023  cf0  8120  fin1a2lem12  8280  mreexexlem3d  13859  efgval  15337  ppttop  17059  0nnei  17164  isarchi  24240  filnetlem4  26347  swrd0  28075  pnonsingN  30569  osumcllem4N  30595
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621
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