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Theorem ncvspfin 4538
 Description: The cardinality of the universe is in the finite Sp set. Theorem X.1.49 of [Rosser] p. 534. (Contributed by SF, 27-Jan-2015.)
Assertion
Ref Expression
ncvspfin Ncfin Spfin

Proof of Theorem ncvspfin
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ncfinex 4472 . . . 4 Ncfin
21elintab 3937 . . 3 Ncfin Ncfin Sfin Ncfin Sfin Ncfin
3 simpl 443 . . 3 Ncfin Sfin Ncfin
42, 3mpgbir 1550 . 2 Ncfin Ncfin Sfin
5 df-spfin 4447 . 2 Spfin Ncfin Sfin
64, 5eleqtrri 2426 1 Ncfin Spfin
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 358  wal 1540   wcel 1710  cab 2339  wral 2614  cvv 2859  cint 3926   Ncfin cncfin 4434   Sfin wsfin 4438   Spfin cspfin 4439 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-int 3927  df-iota 4339  df-ncfin 4442  df-spfin 4447 This theorem is referenced by:  spfininduct  4540  1cspfin  4543  vfinspss  4551  vfinspeqtncv  4553
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