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Theorem ss0 3257
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 3256 . 2 (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
21biimpi 113 1 (𝐴 ⊆ ∅ → 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wss 2917  c0 3224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225
This theorem is referenced by:  sseq0  3258  abf  3260  eq0rdv  3261  ssdisj  3277  disjpss  3278  0dif  3295  poirr2  4717  iotanul  4882  f00  5081  phplem2  6316  php5dom  6325  ixxdisj  8772  icodisj  8860  ioodisj  8861  uzdisj  8955  nn0disj  8995
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