Theorem ncfintfin 4495

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 4474	. . . 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 4492	. . 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 4302	. . . 4 |- (A e. V -> ~P1 A e. _V)
6		ncfinprop 4474	. . . 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 4494	. . 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 4430	. 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 2859  ~P₁ cpw1 4135   Nn cnnc 4373   Fin cfin 4376   Nc_fin cncfin 4434   T_fin ctfin 4435

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-ncfin 4442  df-tfin 4443

This theorem is referenced by:  tncveqnc1fin  4544