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Theorem ss0b 4358
Description: Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)
Assertion
Ref Expression
ss0b (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)

Proof of Theorem ss0b
StepHypRef Expression
1 0ss 4357 . . 3 ∅ ⊆ 𝐴
2 eqss 3960 . . 3 (𝐴 = ∅ ↔ (𝐴 ⊆ ∅ ∧ ∅ ⊆ 𝐴))
31, 2mpbiran2 709 . 2 (𝐴 = ∅ ↔ 𝐴 ⊆ ∅)
43bicomi 223 1 (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wss 3911  c0 4283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-dif 3914  df-in 3918  df-ss 3928  df-nul 4284
This theorem is referenced by:  ss0  4359  un00  4403  pw0  4773  al0ssb  5266  fnsuppeq0  8124  cnfcom2lem  9642  card0  9899  kmlem5  10095  cf0  10192  fin1a2lem12  10352  mreexexlem3d  17531  efgval  19504  ppttop  22373  0nnei  22479  disjunsn  31558  isarchi  32067  filnetlem4  34899  bj-pw0ALT  35566  coss0  36987  pnonsingN  38442  osumcllem4N  38468  resnonrel  41952  ntrneicls11  42450  ntrneikb  42454  sprsymrelfvlem  45768
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