Theorem spfinsfincl 4540

Description: If X is in Sp_fin and Z is smaller than X, then Z is also in Sp_fin. Theorem X.1.50 of [Rosser] p. 534. (Contributed by SF, 27-Jan-2015.)

Assertion

Ref	Expression
spfinsfincl	⊢ ((X ∈ Spfin ∧ Sfin (Z, X)) → Z ∈ Spfin )

Proof of Theorem spfinsfincl

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

Step	Hyp	Ref	Expression
1		df-sfin 4447	. . . 4 ⊢ ( Sfin (Z, X) ↔ (Z ∈ Nn ∧ X ∈ Nn ∧ ∃y(℘1y ∈ Z ∧ ℘y ∈ X)))
2		sfineq1 4527	. . . . . . 7 ⊢ (z = Z → ( Sfin (z, x) ↔ Sfin (Z, x)))
3		eleq1 2413	. . . . . . . 8 ⊢ (z = Z → (z ∈ Spfin ↔ Z ∈ Spfin ))
4	3	imbi2d 307	. . . . . . 7 ⊢ (z = Z → ((x ∈ Spfin → z ∈ Spfin ) ↔ (x ∈ Spfin → Z ∈ Spfin )))
5	2, 4	imbi12d 311	. . . . . 6 ⊢ (z = Z → (( Sfin (z, x) → (x ∈ Spfin → z ∈ Spfin )) ↔ ( Sfin (Z, x) → (x ∈ Spfin → Z ∈ Spfin ))))
6		sfineq2 4528	. . . . . . 7 ⊢ (x = X → ( Sfin (Z, x) ↔ Sfin (Z, X)))
7		eleq1 2413	. . . . . . . 8 ⊢ (x = X → (x ∈ Spfin ↔ X ∈ Spfin ))
8	7	imbi1d 308	. . . . . . 7 ⊢ (x = X → ((x ∈ Spfin → Z ∈ Spfin ) ↔ (X ∈ Spfin → Z ∈ Spfin )))
9	6, 8	imbi12d 311	. . . . . 6 ⊢ (x = X → (( Sfin (Z, x) → (x ∈ Spfin → Z ∈ Spfin )) ↔ ( Sfin (Z, X) → (X ∈ Spfin → Z ∈ Spfin ))))
10		sfineq2 4528	. . . . . . . . . . . . . . 15 ⊢ (p = x → ( Sfin (q, p) ↔ Sfin (q, x)))
11	10	imbi1d 308	. . . . . . . . . . . . . 14 ⊢ (p = x → (( Sfin (q, p) → q ∈ a) ↔ ( Sfin (q, x) → q ∈ a)))
12	11	albidv 1625	. . . . . . . . . . . . 13 ⊢ (p = x → (∀q( Sfin (q, p) → q ∈ a) ↔ ∀q( Sfin (q, x) → q ∈ a)))
13	12	rspcv 2952	. . . . . . . . . . . 12 ⊢ (x ∈ a → (∀p ∈ a ∀q( Sfin (q, p) → q ∈ a) → ∀q( Sfin (q, x) → q ∈ a)))
14		sfineq1 4527	. . . . . . . . . . . . . . 15 ⊢ (q = z → ( Sfin (q, x) ↔ Sfin (z, x)))
15		eleq1 2413	. . . . . . . . . . . . . . 15 ⊢ (q = z → (q ∈ a ↔ z ∈ a))
16	14, 15	imbi12d 311	. . . . . . . . . . . . . 14 ⊢ (q = z → (( Sfin (q, x) → q ∈ a) ↔ ( Sfin (z, x) → z ∈ a)))
17	16	spv 1998	. . . . . . . . . . . . 13 ⊢ (∀q( Sfin (q, x) → q ∈ a) → ( Sfin (z, x) → z ∈ a))
18	17	com12 27	. . . . . . . . . . . 12 ⊢ ( Sfin (z, x) → (∀q( Sfin (q, x) → q ∈ a) → z ∈ a))
19	13, 18	syl9r 67	. . . . . . . . . . 11 ⊢ ( Sfin (z, x) → (x ∈ a → (∀p ∈ a ∀q( Sfin (q, p) → q ∈ a) → z ∈ a)))
20	19	com23 72	. . . . . . . . . 10 ⊢ ( Sfin (z, x) → (∀p ∈ a ∀q( Sfin (q, p) → q ∈ a) → (x ∈ a → z ∈ a)))
21	20	adantld 453	. . . . . . . . 9 ⊢ ( Sfin (z, x) → (( Ncfin V ∈ a ∧ ∀p ∈ a ∀q( Sfin (q, p) → q ∈ a)) → (x ∈ a → z ∈ a)))
22	21	a2d 23	. . . . . . . 8 ⊢ ( Sfin (z, x) → ((( Ncfin V ∈ a ∧ ∀p ∈ a ∀q( Sfin (q, p) → q ∈ a)) → x ∈ a) → (( Ncfin V ∈ a ∧ ∀p ∈ a ∀q( Sfin (q, p) → q ∈ a)) → z ∈ a)))
23	22	alimdv 1621	. . . . . . 7 ⊢ ( Sfin (z, x) → (∀a(( Ncfin V ∈ a ∧ ∀p ∈ a ∀q( Sfin (q, p) → q ∈ a)) → x ∈ a) → ∀a(( Ncfin V ∈ a ∧ ∀p ∈ a ∀q( Sfin (q, p) → q ∈ a)) → z ∈ a)))
24		df-spfin 4448	. . . . . . . . 9 ⊢ Spfin = ∩{a ∣ ( Ncfin V ∈ a ∧ ∀p ∈ a ∀q( Sfin (q, p) → q ∈ a))}
25	24	eleq2i 2417	. . . . . . . 8 ⊢ (x ∈ Spfin ↔ x ∈ ∩{a ∣ ( Ncfin V ∈ a ∧ ∀p ∈ a ∀q( Sfin (q, p) → q ∈ a))})
26		vex 2863	. . . . . . . . 9 ⊢ x ∈ V
27	26	elintab 3938	. . . . . . . 8 ⊢ (x ∈ ∩{a ∣ ( Ncfin V ∈ a ∧ ∀p ∈ a ∀q( Sfin (q, p) → q ∈ a))} ↔ ∀a(( Ncfin V ∈ a ∧ ∀p ∈ a ∀q( Sfin (q, p) → q ∈ a)) → x ∈ a))
28	25, 27	bitri 240	. . . . . . 7 ⊢ (x ∈ Spfin ↔ ∀a(( Ncfin V ∈ a ∧ ∀p ∈ a ∀q( Sfin (q, p) → q ∈ a)) → x ∈ a))
29	24	eleq2i 2417	. . . . . . . 8 ⊢ (z ∈ Spfin ↔ z ∈ ∩{a ∣ ( Ncfin V ∈ a ∧ ∀p ∈ a ∀q( Sfin (q, p) → q ∈ a))})
30		vex 2863	. . . . . . . . 9 ⊢ z ∈ V
31	30	elintab 3938	. . . . . . . 8 ⊢ (z ∈ ∩{a ∣ ( Ncfin V ∈ a ∧ ∀p ∈ a ∀q( Sfin (q, p) → q ∈ a))} ↔ ∀a(( Ncfin V ∈ a ∧ ∀p ∈ a ∀q( Sfin (q, p) → q ∈ a)) → z ∈ a))
32	29, 31	bitri 240	. . . . . . 7 ⊢ (z ∈ Spfin ↔ ∀a(( Ncfin V ∈ a ∧ ∀p ∈ a ∀q( Sfin (q, p) → q ∈ a)) → z ∈ a))
33	23, 28, 32	3imtr4g 261	. . . . . 6 ⊢ ( Sfin (z, x) → (x ∈ Spfin → z ∈ Spfin ))
34	5, 9, 33	vtocl2g 2919	. . . . 5 ⊢ ((Z ∈ Nn ∧ X ∈ Nn ) → ( Sfin (Z, X) → (X ∈ Spfin → Z ∈ Spfin )))
35	34	3adant3 975	. . . 4 ⊢ ((Z ∈ Nn ∧ X ∈ Nn ∧ ∃y(℘1y ∈ Z ∧ ℘y ∈ X)) → ( Sfin (Z, X) → (X ∈ Spfin → Z ∈ Spfin )))
36	1, 35	sylbi 187	. . 3 ⊢ ( Sfin (Z, X) → ( Sfin (Z, X) → (X ∈ Spfin → Z ∈ Spfin )))
37	36	pm2.43i 43	. 2 ⊢ ( Sfin (Z, X) → (X ∈ Spfin → Z ∈ Spfin ))
38	37	impcom 419	1 ⊢ ((X ∈ Spfin ∧ Sfin (Z, X)) → Z ∈ Spfin )

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  Vcvv 2860  ℘cpw 3723  ∩cint 3927  ℘₁cpw1 4136   Nn cnnc 4374   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

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-int 3928  df-sfin 4447  df-spfin 4448

This theorem is referenced by:  spfininduct  4541  1cspfin  4544  vfinspsslem1  4551  vfinspclt  4553