Theorem tfinex 4486

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 4444	. 2 ⊢ Tfin A = if(A = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x)))
2		0ex 4111	. . 3 ⊢ ∅ ∈ V
3		iotaex 4357	. . 3 ⊢ (℩x(x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x)) ∈ V
4	2, 3	ifex 3721	. 2 ⊢ if(A = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x))) ∈ V
5	1, 4	eqeltri 2423	1 ⊢ Tfin A ∈ V

This proof depends on syntax axioms:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860  ∅c0 3551   ifcif 3663  ℘₁cpw1 4136  ℩cio 4338   Nn cnnc 4374   T_fin ctfin 4436

This proof depends on 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  ax-nin 4079

This proof 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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-if 3664  df-sn 3742  df-pr 3743  df-uni 3893  df-iota 4340  df-tfin 4444

This theorem is used by:  tfinltfinlem1  4501  tfinltfin  4502  eventfin  4518  oddtfin  4519  sfintfinlem1  4532  tfinnnlem1  4534  tfinnn  4535  vfinspsslem1  4551