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

Theorem tfinnul 4491
 Description: The finite T operator applied to the empty set is empty. Theorem X.1.29 of [Rosser] p. 528. (Contributed by SF, 22-Jan-2015.)
Assertion
Ref Expression
tfinnul Tfin =

Proof of Theorem tfinnul
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-tfin 4443 . 2 Tfin = if( = , , (℩x(x Nn y 1y x)))
2 eqid 2353 . . 3 =
3 iftrue 3668 . . 3 ( = → if( = , , (℩x(x Nn y 1y x))) = )
42, 3ax-mp 5 . 2 if( = , , (℩x(x Nn y 1y x))) =
51, 4eqtri 2373 1 Tfin =
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ∅c0 3550   ifcif 3662  ℘1cpw1 4135  ℩cio 4337   Nn cnnc 4373   Tfin ctfin 4435 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3663  df-tfin 4443 This theorem is referenced by:  tfincl  4492  tfin11  4493  tfinltfinlem1  4500  tfinltfin  4501  vfinncvntnn  4548
 Copyright terms: Public domain W3C validator