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Theorem ncfinex 4472
 Description: The finite cardinality of a set exists. (Contributed by SF, 27-Jan-2015.)
Assertion
Ref Expression
ncfinex Ncfin A V

Proof of Theorem ncfinex
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 df-ncfin 4442 . 2 Ncfin A = (℩x(x Nn A x))
2 iotaex 4356 . 2 (℩x(x Nn A x)) V
31, 2eqeltri 2423 1 Ncfin A V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   ∈ wcel 1710  Vcvv 2859  ℩cio 4337   Nn cnnc 4373   Ncfin cncfin 4434 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-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
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