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.

Step	Hyp	Ref	Expression
1		df-tfin 4443	. 2 |- Tfin (/) = if((/) = (/), (/), (iotax(x e. Nn /\ E.y e. (/) ~P1 y e. x)))
2		eqid 2353	. . 3 |- (/) = (/)
3		iftrue 3668	. . 3 |- ((/) = (/) -> if((/) = (/), (/), (iotax(x e. Nn /\ E.y e. (/) ~P1 y e. x))) = (/))
4	2, 3	ax-mp 5	. 2 |- if((/) = (/), (/), (iotax(x e. Nn /\ E.y e. (/) ~P1 y e. x))) = (/)
5	1, 4	eqtri 2373	1 |- Tfin (/) = (/)

Syntax hints:   /\ wa 358   = wceq 1642   e. wcel 1710  E.wrex 2615  (/)c0 3550  ifcif 3662  ~P₁ cpw1 4135  iotacio 4337   Nn cnnc 4373   T_fin 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