Theorem ss0b 4354

Description: Any subset of the empty set is empty. Dual of vss 4361. 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 4353	. . 3 ⊢ ∅ ⊆ 𝐴
2		eqss 3928	. . 3 ⊢ (𝐴 = ∅ ↔ (𝐴 ⊆ ∅ ∧ ∅ ⊆ 𝐴))
3	1, 2	mpbiran2 723	. 2 ⊢ (𝐴 = ∅ ↔ 𝐴 ⊆ ∅)
4	3	bicomi 227	1 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ⊆ wss 3900  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ss 3901  df-dif 4050  df-nul 4282

This theorem is used by:  ss0  4355  sseq0b  4356  un00  4360  pw0  4835  al0ssb  5268  fnsuppeq0  8247  cnfcom2lem  9739  card0  10076  kmlem5  10272  cf0  10367  fin1a2lem12  10528  mreexexlem3d  17870  efgval  19982  ppttop  23380  0nnei  23485  bdayfinbndlem2  28911  disjunsn  33286  isarchi  33842  0elscott  35878  filnetlem4  37384  bj-pw0ALT  38234  coss0  39767  pnonsingN  41256  osumcllem4N  41282  resnonrel  44853  ntrneicls11  45351  ntrneikb  45355  sprsymrelfvlem  48845  isubgr0uhgr  49244  iuneq0  50261