Theorem ncfinex 4472

Description: The finite cardinality of a set exists. (Contributed by SF, 27-Jan-2015.)

Assertion

Ref	Expression
ncfinex	|- Ncfin A e. _V

Proof of Theorem ncfinex

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ncfin 4442	. 2 |- Ncfin A = (iotax(x e. Nn /\ A e. x))
2		iotaex 4356	. 2 |- (iotax(x e. Nn /\ A e. x)) e. _V
3	1, 2	eqeltri 2423	1 |- Ncfin A e. _V

Syntax hints:   /\ wa 358   e. wcel 1710  _Vcvv 2859  iotacio 4337   Nn cnnc 4373   Nc_fin cncfin 4434

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  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-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339  df-ncfin 4442

This theorem is referenced by:  ncvspfin  4538  vfintle  4546  vfin1cltv  4547  vfinncvntsp  4549  vfinspss  4551  vfinspclt  4552  vfinspeqtncv  4553  vfinncsp  4554