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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 Fin A V) → Tfin Ncfin A = Ncfin 1A)

Proof of Theorem ncfintfin
StepHypRef Expression
1 ncfinprop 4474 . . . 4 ((V Fin A V) → ( Ncfin A Nn A Ncfin A))
21simpld 445 . . 3 ((V Fin A V) → Ncfin A Nn )
3 tfincl 4492 . . 3 ( Ncfin A NnTfin Ncfin A Nn )
42, 3syl 15 . 2 ((V Fin A V) → Tfin Ncfin A Nn )
5 pw1exg 4302 . . . 4 (A V1A V)
6 ncfinprop 4474 . . . 4 ((V Fin 1A V) → ( Ncfin 1A Nn 1A Ncfin 1A))
75, 6sylan2 460 . . 3 ((V Fin A V) → ( Ncfin 1A Nn 1A Ncfin 1A))
87simpld 445 . 2 ((V Fin A V) → Ncfin 1A Nn )
9 tfinpw1 4494 . . 3 (( Ncfin A Nn A Ncfin A) → 1A Tfin Ncfin A)
101, 9syl 15 . 2 ((V Fin A V) → 1A Tfin Ncfin A)
117simprd 449 . 2 ((V Fin A V) → 1A Ncfin 1A)
12 nnceleq 4430 . 2 ((( Tfin Ncfin A Nn Ncfin 1A Nn ) (1A Tfin Ncfin A 1A Ncfin 1A)) → Tfin Ncfin A = Ncfin 1A)
134, 8, 10, 11, 12syl22anc 1183 1 ((V Fin A V) → Tfin Ncfin A = Ncfin 1A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ℘1cpw1 4135   Nn cnnc 4373   Fin cfin 4376   Ncfin cncfin 4434   Tfin 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
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