Theorem ss0b 4401

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 4400	. . 3 ⊢ ∅ ⊆ 𝐴
2		eqss 3999	. . 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 3951  ∅c0 4333

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-dif 3954  df-ss 3968  df-nul 4334

This theorem is referenced by:  ss0  4402  un00  4445  pw0  4812  al0ssb  5308  fnsuppeq0  8217  cnfcom2lem  9741  card0  9998  kmlem5  10195  cf0  10291  fin1a2lem12  10451  mreexexlem3d  17689  efgval  19735  ppttop  23014  0nnei  23120  disjunsn  32607  isarchi  33189  filnetlem4  36382  bj-pw0ALT  37050  coss0  38480  pnonsingN  39935  osumcllem4N  39961  resnonrel  43605  ntrneicls11  44103  ntrneikb  44107  sprsymrelfvlem  47477  isubgr0uhgr  47859