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Theorem 1cnnc 4408
Description: Cardinal one is a finite cardinal. Theorem X.1.12 of [Rosser] p. 277. (Contributed by SF, 16-Jan-2015.)
Assertion
Ref Expression
1cnnc 1c Nn

Proof of Theorem 1cnnc
StepHypRef Expression
1 addcid1 4405 . . 3 (1c +c 0c) = 1c
2 addccom 4406 . . 3 (1c +c 0c) = (0c +c 1c)
31, 2eqtr3i 2375 . 2 1c = (0c +c 1c)
4 peano1 4402 . . 3 0c Nn
5 peano2 4403 . . 3 (0c Nn → (0c +c 1c) Nn )
64, 5ax-mp 8 . 2 (0c +c 1c) Nn
73, 6eqeltri 2423 1 1c Nn
Colors of variables: wff setvar class
Syntax hints:   wcel 1710  1cc1c 4134   Nn cnnc 4373  0cc0c 4374   +c cplc 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-pw 3724  df-sn 3741  df-pr 3742  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193  df-0c 4377  df-addc 4378  df-nnc 4379
This theorem is referenced by:  snfi  4431  ncfinsn  4476  oddtfin  4518  nnpweq  4523  sfin01  4528  2nnc  6167  nnc3n3p1  6278  nchoicelem12  6300  nchoicelem17  6305
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