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Theorem ss0 3434
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 3433 . 2  |-  ( A 
C_  (/)  <->  A  =  (/) )
21biimpi 119 1  |-  ( A 
C_  (/)  ->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335    C_ wss 3102   (/)c0 3394
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395
This theorem is referenced by:  sseq0  3435  abf  3437  eq0rdv  3438  ssdisj  3450  0dif  3465  poirr2  4978  iotanul  5150  f00  5361  map0b  6632  phplem2  6798  php5dom  6808  sbthlem7  6907  fi0  6919  casefun  7029  caseinj  7033  djufun  7048  djuinj  7050  nnnninfeq  7071  exmidomni  7085  ixxdisj  9807  icodisj  9896  ioodisj  9897  uzdisj  9995  nn0disj  10037  fsum2dlemstep  11331  fprodssdc  11487  fprod2dlemstep  11519  ntrcls0  12531
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