Theorem ncvspfin 4539

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 4473	. . . 4 ⊢ Ncfin V ∈ V
2	1	elintab 3938	. . 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 4448	. 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 2615  Vcvv 2860  ∩cint 3927   Nc_fin cncfin 4435   S_fin wsfin 4439   Sp_fin cspfin 4440

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-int 3928  df-iota 4340  df-ncfin 4443  df-spfin 4448

This theorem is referenced by:  spfininduct  4541  1cspfin  4544  vfinspss  4552  vfinspeqtncv  4554