Theorem vfinspnn 4542

Description: If the universe is finite, then Sp_fin is a subset of the nonempty naturals. Theorem X.1.53 of [Rosser] p. 534. (Contributed by SF, 27-Jan-2015.)

Assertion

Ref	Expression
vfinspnn	|- (_V e. Fin -> Spfin C_ ( Nn \ {(/)}))

Proof of Theorem vfinspnn

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

Step	Hyp	Ref	Expression
1		vvex 4110	. . . . 5 |- _V e. _V
2		ncfinprop 4475	. . . . 5 |- ((_V e. Fin /\ _V e. _V) -> ( Ncfin _V e. Nn /\ _V e. Ncfin _V))
3	1, 2	mpan2 652	. . . 4 |- (_V e. Fin -> ( Ncfin _V e. Nn /\ _V e. Ncfin _V))
4		ne0i 3557	. . . . 5 |- (_V e. Ncfin _V -> Ncfin _V =/= (/))
5	4	anim2i 552	. . . 4 |- (( Ncfin _V e. Nn /\ _V e. Ncfin _V) -> ( Ncfin _V e. Nn /\ Ncfin _V =/= (/)))
6	3, 5	syl 15	. . 3 |- (_V e. Fin -> ( Ncfin _V e. Nn /\ Ncfin _V =/= (/)))
7		eldifsn 3840	. . 3 |- ( Ncfin _V e. ( Nn \ {(/)}) <-> ( Ncfin _V e. Nn /\ Ncfin _V =/= (/)))
8	6, 7	sylibr 203	. 2 |- (_V e. Fin -> Ncfin _V e. ( Nn \ {(/)}))
9		df-sfin 4447	. . . . . 6 |- ( Sfin (z, x) <-> (z e. Nn /\ x e. Nn /\ E.y(~P1 y e. z /\ ~Py e. x)))
10		ne0i 3557	. . . . . . . . . 10 |- (~P1 y e. z -> z =/= (/))
11	10	adantr 451	. . . . . . . . 9 |- ((~P1 y e. z /\ ~Py e. x) -> z =/= (/))
12	11	exlimiv 1634	. . . . . . . 8 |- (E.y(~P1 y e. z /\ ~Py e. x) -> z =/= (/))
13		eldifsn 3840	. . . . . . . . 9 |- (z e. ( Nn \ {(/)}) <-> (z e. Nn /\ z =/= (/)))
14	13	biimpri 197	. . . . . . . 8 |- ((z e. Nn /\ z =/= (/)) -> z e. ( Nn \ {(/)}))
15	12, 14	sylan2 460	. . . . . . 7 |- ((z e. Nn /\ E.y(~P1 y e. z /\ ~Py e. x)) -> z e. ( Nn \ {(/)}))
16	15	3adant2 974	. . . . . 6 |- ((z e. Nn /\ x e. Nn /\ E.y(~P1 y e. z /\ ~Py e. x)) -> z e. ( Nn \ {(/)}))
17	9, 16	sylbi 187	. . . . 5 |- ( Sfin (z, x) -> z e. ( Nn \ {(/)}))
18	17	adantl 452	. . . 4 |- ((x e. ( Nn \ {(/)}) /\ Sfin (z, x)) -> z e. ( Nn \ {(/)}))
19	18	ax-gen 1546	. . 3 |- A.z((x e. ( Nn \ {(/)}) /\ Sfin (z, x)) -> z e. ( Nn \ {(/)}))
20	19	rgenw 2682	. 2 |- A.x e. Spfin A.z((x e. ( Nn \ {(/)}) /\ Sfin (z, x)) -> z e. ( Nn \ {(/)}))
21		nncex 4397	. . . 4 |- Nn e. _V
22		snex 4112	. . . 4 |- {(/)} e. _V
23	21, 22	difex 4108	. . 3 |- ( Nn \ {(/)}) e. _V
24		spfininduct 4541	. . 3 |- ((( Nn \ {(/)}) e. _V /\ Ncfin _V e. ( Nn \ {(/)}) /\ A.x e. Spfin A.z((x e. ( Nn \ {(/)}) /\ Sfin (z, x)) -> z e. ( Nn \ {(/)}))) -> Spfin C_ ( Nn \ {(/)}))
25	23, 24	mp3an1 1264	. 2 |- (( Ncfin _V e. ( Nn \ {(/)}) /\ A.x e. Spfin A.z((x e. ( Nn \ {(/)}) /\ Sfin (z, x)) -> z e. ( Nn \ {(/)}))) -> Spfin C_ ( Nn \ {(/)}))
26	8, 20, 25	sylancl 643	1 |- (_V e. Fin -> Spfin C_ ( Nn \ {(/)}))

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934  A.wal 1540  E.wex 1541   e. wcel 1710   =/= wne 2517  A.wral 2615  _Vcvv 2860   \ cdif 3207   C_ wss 3258  (/)c0 3551  ~Pcpw 3723  {csn 3738  ~P₁ cpw1 4136   Nn cnnc 4374   Fin cfin 4377   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-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-ncfin 4443  df-sfin 4447  df-spfin 4448

This theorem is referenced by:  vfinncvntsp  4550  vfinspsslem1  4551  vfinncsp  4555