Theorem tfinex 4485

Description: The finite T operator is always a set. (Contributed by SF, 26-Jan-2015.)

Assertion

Ref	Expression
tfinex	⊢ Tfin A ∈ V

Proof of Theorem tfinex

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-tfin 4443	. 2 ⊢ Tfin A = if(A = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x)))
2		0ex 4110	. . 3 ⊢ ∅ ∈ V
3		iotaex 4356	. . 3 ⊢ (℩x(x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x)) ∈ V
4	2, 3	ifex 3720	. 2 ⊢ if(A = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x))) ∈ V
5	1, 4	eqeltri 2423	1 ⊢ Tfin A ∈ V

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859  ∅c0 3550   ifcif 3662  ℘₁cpw1 4135  ℩cio 4337   Nn cnnc 4373   T_fin ctfin 4435

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  ax-nin 4078

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-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-if 3663  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339  df-tfin 4443

This theorem is referenced by:  tfinltfinlem1  4500  tfinltfin  4501  eventfin  4517  oddtfin  4518  sfintfinlem1  4531  tfinnnlem1  4533  tfinnn  4534  vfinspsslem1  4550