ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ss0 GIF version

Theorem ss0 3350
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
StepHypRef Expression
1 ss0b 3349 . 2 (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
21biimpi 119 1 (𝐴 ⊆ ∅ → 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1299  wss 3021  c0 3310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-dif 3023  df-in 3027  df-ss 3034  df-nul 3311
This theorem is referenced by:  sseq0  3351  abf  3353  eq0rdv  3354  ssdisj  3366  0dif  3381  poirr2  4867  iotanul  5039  f00  5250  map0b  6511  phplem2  6676  php5dom  6686  sbthlem7  6779  casefun  6885  caseinj  6889  djufun  6904  djuinj  6906  exmidomni  6926  ixxdisj  9527  icodisj  9616  ioodisj  9617  uzdisj  9714  nn0disj  9756  fsum2dlemstep  11042  ntrcls0  12082  nninfalllemn  12786
  Copyright terms: Public domain W3C validator