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Theorem ss0 3323
Description: Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
Assertion
Ref Expression
ss0  |-  ( A 
C_  (/)  ->  A  =  (/) )

Proof of Theorem ss0
StepHypRef Expression
1 ss0b 3322 . 2  |-  ( A 
C_  (/)  <->  A  =  (/) )
21biimpi 118 1  |-  ( A 
C_  (/)  ->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    C_ wss 2999   (/)c0 3286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-nul 3287
This theorem is referenced by:  sseq0  3324  abf  3326  eq0rdv  3327  ssdisj  3339  0dif  3354  poirr2  4824  iotanul  4995  f00  5202  map0b  6444  phplem2  6569  php5dom  6579  sbthlem7  6672  casefun  6776  caseinj  6780  djufun  6784  djuinj  6786  exmidomni  6798  ixxdisj  9321  icodisj  9409  ioodisj  9410  uzdisj  9507  nn0disj  9549  fsum2dlemstep  10828  nninfalllemn  11898
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