Theorem ss0b 4346

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

Step	Hyp	Ref	Expression
1		0ss 4345	. . 3 ⊢ ∅ ⊆ 𝐴
2		eqss 3945	. . 3 ⊢ (𝐴 = ∅ ↔ (𝐴 ⊆ ∅ ∧ ∅ ⊆ 𝐴))
3	1, 2	mpbiran2 710	. 2 ⊢ (𝐴 = ∅ ↔ 𝐴 ⊆ ∅)
4	3	bicomi 224	1 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 206   = wceq 1541   ⊆ wss 3897  ∅c0 4278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-dif 3900  df-ss 3914  df-nul 4279

This theorem is referenced by:  ss0  4347  un00  4390  pw0  4759  al0ssb  5241  fnsuppeq0  8117  cnfcom2lem  9586  card0  9846  kmlem5  10041  cf0  10137  fin1a2lem12  10297  mreexexlem3d  17547  efgval  19624  ppttop  22917  0nnei  23022  disjunsn  32566  isarchi  33143  filnetlem4  36415  bj-pw0ALT  37083  coss0  38516  pnonsingN  39972  osumcllem4N  39998  resnonrel  43625  ntrneicls11  44123  ntrneikb  44127  sprsymrelfvlem  47521  isubgr0uhgr  47904  iuneq0  48850