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Theorem ss0 3308
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 3307 . 2 (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
21biimpi 118 1 (𝐴 ⊆ ∅ → 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wss 2986  c0 3272
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-dif 2988  df-in 2992  df-ss 2999  df-nul 3273
This theorem is referenced by:  sseq0  3309  abf  3311  eq0rdv  3312  ssdisj  3324  0dif  3339  poirr2  4782  iotanul  4952  f00  5153  map0b  6377  phplem2  6502  php5dom  6512  sbthlem7  6593  casefun  6697  caseinj  6701  djufun  6705  djuinj  6707  exmidomni  6719  ixxdisj  9230  icodisj  9318  ioodisj  9319  uzdisj  9414  nn0disj  9453  nninfalllemn  11254
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