New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  tfin1c GIF version

Theorem tfin1c 4499
 Description: The finite T operator is idempotent over 1c. Theorem X.1.34(a) of [Rosser] p. 529. (Contributed by SF, 30-Jan-2015.)
Assertion
Ref Expression
tfin1c Tfin 1c = 1c

Proof of Theorem tfin1c
StepHypRef Expression
1 peano1 4402 . . 3 0c Nn
2 addcid2 4407 . . . 4 (0c +c 1c) = 1c
3 1cex 4142 . . . . . 6 1c V
43snel1c 4140 . . . . 5 {1c} 1c
5 ne0i 3556 . . . . 5 ({1c} 1c → 1c)
64, 5ax-mp 5 . . . 4 1c
72, 6eqnetri 2533 . . 3 (0c +c 1c) ≠
8 tfinsuc 4498 . . 3 ((0c Nn (0c +c 1c) ≠ ) → Tfin (0c +c 1c) = ( Tfin 0c +c 1c))
91, 7, 8mp2an 653 . 2 Tfin (0c +c 1c) = ( Tfin 0c +c 1c)
10 tfineq 4488 . . 3 ((0c +c 1c) = 1cTfin (0c +c 1c) = Tfin 1c)
112, 10ax-mp 5 . 2 Tfin (0c +c 1c) = Tfin 1c
12 tfin0c 4497 . . . 4 Tfin 0c = 0c
1312addceq1i 4386 . . 3 ( Tfin 0c +c 1c) = (0c +c 1c)
1413, 2eqtri 2373 . 2 ( Tfin 0c +c 1c) = 1c
159, 11, 143eqtr3i 2381 1 Tfin 1c = 1c
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∅c0 3550  {csn 3737  1cc1c 4134   Nn cnnc 4373  0cc0c 4374   +c cplc 4375   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-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-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-tfin 4443 This theorem is referenced by:  oddtfin  4518  sfintfin  4532
 Copyright terms: Public domain W3C validator