Theorem ss0b 4398

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 4397	. . 3 ⊢ ∅ ⊆ 𝐴
2		eqss 3998	. . 3 ⊢ (𝐴 = ∅ ↔ (𝐴 ⊆ ∅ ∧ ∅ ⊆ 𝐴))
3	1, 2	mpbiran2 709	. 2 ⊢ (𝐴 = ∅ ↔ 𝐴 ⊆ ∅)
4	3	bicomi 223	1 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 205   = wceq 1542   ⊆ wss 3949  ∅c0 4323

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324

This theorem is referenced by:  ss0  4399  un00  4443  pw0  4816  al0ssb  5309  fnsuppeq0  8177  cnfcom2lem  9696  card0  9953  kmlem5  10149  cf0  10246  fin1a2lem12  10406  mreexexlem3d  17590  efgval  19585  ppttop  22510  0nnei  22616  disjunsn  31825  isarchi  32328  filnetlem4  35266  bj-pw0ALT  35930  coss0  37349  pnonsingN  38804  osumcllem4N  38830  resnonrel  42343  ntrneicls11  42841  ntrneikb  42845  sprsymrelfvlem  46158