Theorem tfinnul 4492

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 4444	. 2 ⊢ Tfin ∅ = if(∅ = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ ∅ ℘1y ∈ x)))
2		eqid 2353	. . 3 ⊢ ∅ = ∅
3		iftrue 3669	. . 3 ⊢ (∅ = ∅ → if(∅ = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ ∅ ℘1y ∈ x))) = ∅)
4	2, 3	ax-mp 5	. 2 ⊢ if(∅ = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ ∅ ℘1y ∈ x))) = ∅
5	1, 4	eqtri 2373	1 ⊢ Tfin ∅ = ∅

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  ∅c0 3551   ifcif 3663  ℘₁cpw1 4136  ℩cio 4338   Nn cnnc 4374   T_fin ctfin 4436

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 3664  df-tfin 4444

This theorem is referenced by:  tfincl  4493  tfin11  4494  tfinltfinlem1  4501  tfinltfin  4502  vfinncvntnn  4549