Theorem ss0b 4381

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 4380	. . 3 ⊢ ∅ ⊆ 𝐴
2		eqss 3979	. . 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 3931  ∅c0 4313

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 2708

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 2715  df-cleq 2728  df-clel 2810  df-dif 3934  df-ss 3948  df-nul 4314

This theorem is referenced by:  ss0  4382  un00  4425  pw0  4793  al0ssb  5283  fnsuppeq0  8196  cnfcom2lem  9720  card0  9977  kmlem5  10174  cf0  10270  fin1a2lem12  10430  mreexexlem3d  17663  efgval  19703  ppttop  22950  0nnei  23055  disjunsn  32580  isarchi  33185  filnetlem4  36404  bj-pw0ALT  37072  coss0  38502  pnonsingN  39957  osumcllem4N  39983  resnonrel  43583  ntrneicls11  44081  ntrneikb  44085  sprsymrelfvlem  47471  isubgr0uhgr  47853  iuneq0  48764