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Theorem ss0 3455
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 3454 . 2  |-  ( A 
C_  (/)  <->  A  =  (/) )
21biimpi 119 1  |-  ( A 
C_  (/)  ->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    C_ wss 3121   (/)c0 3414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415
This theorem is referenced by:  sseq0  3456  abf  3458  eq0rdv  3459  ssdisj  3471  0dif  3486  poirr2  5003  iotanul  5175  f00  5389  map0b  6665  phplem2  6831  php5dom  6841  sbthlem7  6940  fi0  6952  casefun  7062  caseinj  7066  djufun  7081  djuinj  7083  nnnninfeq  7104  exmidomni  7118  ixxdisj  9860  icodisj  9949  ioodisj  9950  uzdisj  10049  nn0disj  10094  fsum2dlemstep  11397  fprodssdc  11553  fprod2dlemstep  11585  ntrcls0  12925
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