Theorem ss0b 4366

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 4365	. . 3 ⊢ ∅ ⊆ 𝐴
2		eqss 3964	. . 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 3916  ∅c0 4298

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-dif 3919  df-ss 3933  df-nul 4299

This theorem is referenced by:  ss0  4367  un00  4410  pw0  4778  al0ssb  5265  fnsuppeq0  8173  cnfcom2lem  9660  card0  9917  kmlem5  10114  cf0  10210  fin1a2lem12  10370  mreexexlem3d  17613  efgval  19653  ppttop  22900  0nnei  23005  disjunsn  32529  isarchi  33142  filnetlem4  36364  bj-pw0ALT  37032  coss0  38465  pnonsingN  39922  osumcllem4N  39948  resnonrel  43574  ntrneicls11  44072  ntrneikb  44076  sprsymrelfvlem  47481  isubgr0uhgr  47863  iuneq0  48797