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Theorem ss0b 4006
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 4005 . . 3 ∅ ⊆ 𝐴
2 eqss 3651 . . 3 (𝐴 = ∅ ↔ (𝐴 ⊆ ∅ ∧ ∅ ⊆ 𝐴))
31, 2mpbiran2 974 . 2 (𝐴 = ∅ ↔ 𝐴 ⊆ ∅)
43bicomi 214 1 (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wss 3607  c0 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949
This theorem is referenced by:  ss0  4007  un00  4044  ssdisjOLD  4060  pw0  4375  fnsuppeq0  7368  cnfcom2lem  8636  card0  8822  kmlem5  9014  cf0  9111  fin1a2lem12  9271  mreexexlem3d  16353  efgval  18176  ppttop  20859  0nnei  20964  disjunsn  29533  isarchi  29864  filnetlem4  32501  coss0  34369  pnonsingN  35537  osumcllem4N  35563  resnonrel  38215  ntrneicls11  38705  ntrneikb  38709  sprsymrelfvlem  42065
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