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 V ∈ Spfin

Proof of Theorem ncvspfin

Dummy variables x a z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ncfinex 4472	. . . 4 ⊢ Ncfin V ∈ V
2	1	elintab 3937	. . 3 ⊢ ( Ncfin V ∈ ∩{a ∣ ( Ncfin V ∈ a ∧ ∀x ∈ a ∀z( Sfin (z, x) → z ∈ a))} ↔ ∀a(( Ncfin V ∈ a ∧ ∀x ∈ a ∀z( Sfin (z, x) → z ∈ a)) → Ncfin V ∈ a))
3		simpl 443	. . 3 ⊢ (( Ncfin V ∈ a ∧ ∀x ∈ a ∀z( Sfin (z, x) → z ∈ a)) → Ncfin V ∈ a)
4	2, 3	mpgbir 1550	. 2 ⊢ Ncfin V ∈ ∩{a ∣ ( Ncfin V ∈ a ∧ ∀x ∈ a ∀z( Sfin (z, x) → z ∈ a))}
5		df-spfin 4447	. 2 ⊢ Spfin = ∩{a ∣ ( Ncfin V ∈ a ∧ ∀x ∈ a ∀z( Sfin (z, x) → z ∈ a))}
6	4, 5	eleqtrri 2426	1 ⊢ Ncfin V ∈ Spfin

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  {cab 2339  ∀wral 2614  Vcvv 2859  ∩cint 3926   Nc_fin cncfin 4434   S_fin wsfin 4438   Sp_fin 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