Theorem ss0b 4354

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 4353	. . 3 ⊢ ∅ ⊆ 𝐴
2		eqss 3953	. . 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 3905  ∅c0 4286

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 3908  df-ss 3922  df-nul 4287

This theorem is referenced by:  ss0  4355  un00  4398  pw0  4766  al0ssb  5250  fnsuppeq0  8132  cnfcom2lem  9616  card0  9873  kmlem5  10068  cf0  10164  fin1a2lem12  10324  mreexexlem3d  17570  efgval  19614  ppttop  22910  0nnei  23015  disjunsn  32556  isarchi  33137  filnetlem4  36357  bj-pw0ALT  37025  coss0  38458  pnonsingN  39915  osumcllem4N  39941  resnonrel  43568  ntrneicls11  44066  ntrneikb  44070  sprsymrelfvlem  47478  isubgr0uhgr  47861  iuneq0  48807