Theorem ss0b 4364

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 4363	. . 3 ⊢ ∅ ⊆ 𝐴
2		eqss 3962	. . 3 ⊢ (𝐴 = ∅ ↔ (𝐴 ⊆ ∅ ∧ ∅ ⊆ 𝐴))
3	1, 2	mpbiran2 710	. 2 ⊢ (𝐴 = ∅ ↔ 𝐴 ⊆ ∅)
4	3	bicomi 224	1 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 206   = wceq 1540   ⊆ wss 3914  ∅c0 4296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-dif 3917  df-ss 3931  df-nul 4297

This theorem is referenced by:  ss0  4365  un00  4408  pw0  4776  al0ssb  5263  fnsuppeq0  8171  cnfcom2lem  9654  card0  9911  kmlem5  10108  cf0  10204  fin1a2lem12  10364  mreexexlem3d  17607  efgval  19647  ppttop  22894  0nnei  22999  disjunsn  32523  isarchi  33136  filnetlem4  36369  bj-pw0ALT  37037  coss0  38470  pnonsingN  39927  osumcllem4N  39953  resnonrel  43581  ntrneicls11  44079  ntrneikb  44083  sprsymrelfvlem  47491  isubgr0uhgr  47873  iuneq0  48807