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Theorem ss0 3407
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 ⊆ ∅ → 𝐴 = ∅)

Proof of Theorem ss0
StepHypRef Expression
1 ss0b 3406 . 2 (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
21biimpi 119 1 (𝐴 ⊆ ∅ → 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  wss 3075  c0 3367
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3077  df-in 3081  df-ss 3088  df-nul 3368
This theorem is referenced by:  sseq0  3408  abf  3410  eq0rdv  3411  ssdisj  3423  0dif  3438  poirr2  4938  iotanul  5110  f00  5321  map0b  6588  phplem2  6754  php5dom  6764  sbthlem7  6858  fi0  6870  casefun  6977  caseinj  6981  djufun  6996  djuinj  6998  exmidomni  7021  ixxdisj  9715  icodisj  9804  ioodisj  9805  uzdisj  9903  nn0disj  9945  fsum2dlemstep  11234  ntrcls0  12337  nninfalllemn  13375
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