Theorem ss0b 4331

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 4330	. . 3 ⊢ ∅ ⊆ 𝐴
2		eqss 3936	. . 3 ⊢ (𝐴 = ∅ ↔ (𝐴 ⊆ ∅ ∧ ∅ ⊆ 𝐴))
3	1, 2	mpbiran2 707	. 2 ⊢ (𝐴 = ∅ ↔ 𝐴 ⊆ ∅)
4	3	bicomi 223	1 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 205   = wceq 1539   ⊆ wss 3887  ∅c0 4256

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257

This theorem is referenced by:  ss0  4332  un00  4376  pw0  4745  al0ssb  5232  fnsuppeq0  8008  cnfcom2lem  9459  card0  9716  kmlem5  9910  cf0  10007  fin1a2lem12  10167  mreexexlem3d  17355  efgval  19323  ppttop  22157  0nnei  22263  disjunsn  30933  isarchi  31436  filnetlem4  34570  bj-pw0ALT  35222  coss0  36597  pnonsingN  37947  osumcllem4N  37973  resnonrel  41200  ntrneicls11  41700  ntrneikb  41704  sprsymrelfvlem  44942