 New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  ss0b GIF version

Theorem ss0b 3580
 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
StepHypRef Expression
1 0ss 3579 . . 3 A
2 eqss 3287 . . 3 (A = ↔ (A A))
31, 2mpbiran2 885 . 2 (A = A )
43bicomi 193 1 (A A = )
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642   ⊆ wss 3257  ∅c0 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551 This theorem is referenced by:  ss0  3581  un00  3586  ssdisj  3600  pw0  4160  ssfin  4470
 Copyright terms: Public domain W3C validator