Theorem ss0b 4360

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 4359	. . 3 ⊢ ∅ ⊆ 𝐴
2		eqss 3959	. . 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 3911  ∅c0 4292

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 3914  df-ss 3928  df-nul 4293

This theorem is referenced by:  ss0  4361  un00  4404  pw0  4772  al0ssb  5258  fnsuppeq0  8148  cnfcom2lem  9630  card0  9887  kmlem5  10084  cf0  10180  fin1a2lem12  10340  mreexexlem3d  17583  efgval  19623  ppttop  22870  0nnei  22975  disjunsn  32496  isarchi  33109  filnetlem4  36342  bj-pw0ALT  37010  coss0  38443  pnonsingN  39900  osumcllem4N  39926  resnonrel  43554  ntrneicls11  44052  ntrneikb  44056  sprsymrelfvlem  47464  isubgr0uhgr  47846  iuneq0  48780