Theorem ncfinex 4473

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 4443	. 2 |- Ncfin A = (iotax(x e. Nn /\ A e. x))
2		iotaex 4357	. 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 2860  iotacio 4338   Nn cnnc 4374   Nc_fin cncfin 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  ax-nin 4079

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 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-sn 3742  df-pr 3743  df-uni 3893  df-iota 4340  df-ncfin 4443

This theorem is referenced by:  ncvspfin  4539  vfintle  4547  vfin1cltv  4548  vfinncvntsp  4550  vfinspss  4552  vfinspclt  4553  vfinspeqtncv  4554  vfinncsp  4555