Theorem ncfintfin 4496

Description: Relationship between finite T operator and finite Nc operation in a finite universe. Corollary of Theorem X.1.31 of [Rosser] p. 529. (Contributed by SF, 24-Jan-2015.)

Assertion

Ref	Expression
ncfintfin	|- ((_V e. Fin /\ A e. V) -> Tfin Ncfin A = Ncfin ~P1 A)

Proof of Theorem ncfintfin

Step	Hyp	Ref	Expression
1		ncfinprop 4475	. . . 4 |- ((_V e. Fin /\ A e. V) -> ( Ncfin A e. Nn /\ A e. Ncfin A))
2	1	simpld 445	. . 3 |- ((_V e. Fin /\ A e. V) -> Ncfin A e. Nn )
3		tfincl 4493	. . 3 |- ( Ncfin A e. Nn -> Tfin Ncfin A e. Nn )
4	2, 3	syl 15	. 2 |- ((_V e. Fin /\ A e. V) -> Tfin Ncfin A e. Nn )
5		pw1exg 4303	. . . 4 |- (A e. V -> ~P1 A e. _V)
6		ncfinprop 4475	. . . 4 |- ((_V e. Fin /\ ~P1 A e. _V) -> ( Ncfin ~P1 A e. Nn /\ ~P1 A e. Ncfin ~P1 A))
7	5, 6	sylan2 460	. . 3 |- ((_V e. Fin /\ A e. V) -> ( Ncfin ~P1 A e. Nn /\ ~P1 A e. Ncfin ~P1 A))
8	7	simpld 445	. 2 |- ((_V e. Fin /\ A e. V) -> Ncfin ~P1 A e. Nn )
9		tfinpw1 4495	. . 3 |- (( Ncfin A e. Nn /\ A e. Ncfin A) -> ~P1 A e. Tfin Ncfin A)
10	1, 9	syl 15	. 2 |- ((_V e. Fin /\ A e. V) -> ~P1 A e. Tfin Ncfin A)
11	7	simprd 449	. 2 |- ((_V e. Fin /\ A e. V) -> ~P1 A e. Ncfin ~P1 A)
12		nnceleq 4431	. 2 |- ((( Tfin Ncfin A e. Nn /\ Ncfin ~P1 A e. Nn ) /\ (~P1 A e. Tfin Ncfin A /\ ~P1 A e. Ncfin ~P1 A)) -> Tfin Ncfin A = Ncfin ~P1 A)
13	4, 8, 10, 11, 12	syl22anc 1183	1 |- ((_V e. Fin /\ A e. V) -> Tfin Ncfin A = Ncfin ~P1 A)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  _Vcvv 2860  ~P₁ cpw1 4136   Nn cnnc 4374   Fin cfin 4377   Nc_fin cncfin 4435   T_fin ctfin 4436

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-tfin 4444

This theorem is referenced by:  tncveqnc1fin  4545