Theorem ss0 3582

Description: Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)

Assertion

Ref	Expression
ss0	⊢ (A ⊆ ∅ → A = ∅)

Proof of Theorem ss0

Step	Hyp	Ref	Expression
1		ss0b 3581	. 2 ⊢ (A ⊆ ∅ ↔ A = ∅)
2	1	biimpi 186	1 ⊢ (A ⊆ ∅ → A = ∅)

Syntax hints:   → wi 4   = wceq 1642   ⊆ wss 3258  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  sseq0  3583  abf  3585  eq0rdv  3586  ssdisj  3601  disjpss  3602  0dif  3622  f00  5250  map0b  6025