Theorem peano2 4404

Description: The finite cardinals are closed under addition of one. Theorem X.1.5 of [Rosser] p. 276. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
peano2	|- (A e. Nn -> (A +o 1c) e. Nn )

Proof of Theorem peano2

Dummy variables a x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addceq1 4384	. . 3 |- (a = A -> (a +o 1c) = (A +o 1c))
2	1	eleq1d 2419	. 2 |- (a = A -> ((a +o 1c) e. Nn <-> (A +o 1c) e. Nn ))
3		addceq1 4384	. . . . . . . 8 |- (y = a -> (y +o 1c) = (a +o 1c))
4	3	eleq1d 2419	. . . . . . 7 |- (y = a -> ((y +o 1c) e. x <-> (a +o 1c) e. x))
5	4	rspccv 2953	. . . . . 6 |- (A.y e. x (y +o 1c) e. x -> (a e. x -> (a +o 1c) e. x))
6	5	adantl 452	. . . . 5 |- ((0c e. x /\ A.y e. x (y +o 1c) e. x) -> (a e. x -> (a +o 1c) e. x))
7	6	a2i 12	. . . 4 |- (((0c e. x /\ A.y e. x (y +o 1c) e. x) -> a e. x) -> ((0c e. x /\ A.y e. x (y +o 1c) e. x) -> (a +o 1c) e. x))
8	7	alimi 1559	. . 3 |- (A.x((0c e. x /\ A.y e. x (y +o 1c) e. x) -> a e. x) -> A.x((0c e. x /\ A.y e. x (y +o 1c) e. x) -> (a +o 1c) e. x))
9		df-nnc 4380	. . . . 5 |- Nn = |^|{x | (0c e. x /\ A.y e. x (y +o 1c) e. x)}
10	9	eleq2i 2417	. . . 4 |- (a e. Nn <-> a e. |^|{x | (0c e. x /\ A.y e. x (y +o 1c) e. x)})
11		vex 2863	. . . . 5 |- a e. _V
12	11	elintab 3938	. . . 4 |- (a e. |^|{x | (0c e. x /\ A.y e. x (y +o 1c) e. x)} <-> A.x((0c e. x /\ A.y e. x (y +o 1c) e. x) -> a e. x))
13	10, 12	bitri 240	. . 3 |- (a e. Nn <-> A.x((0c e. x /\ A.y e. x (y +o 1c) e. x) -> a e. x))
14	9	eleq2i 2417	. . . 4 |- ((a +o 1c) e. Nn <-> (a +o 1c) e. |^|{x | (0c e. x /\ A.y e. x (y +o 1c) e. x)})
15		1cex 4143	. . . . . 6 |- 1c e. _V
16	11, 15	addcex 4395	. . . . 5 |- (a +o 1c) e. _V
17	16	elintab 3938	. . . 4 |- ((a +o 1c) e. |^|{x | (0c e. x /\ A.y e. x (y +o 1c) e. x)} <-> A.x((0c e. x /\ A.y e. x (y +o 1c) e. x) -> (a +o 1c) e. x))
18	14, 17	bitri 240	. . 3 |- ((a +o 1c) e. Nn <-> A.x((0c e. x /\ A.y e. x (y +o 1c) e. x) -> (a +o 1c) e. x))
19	8, 13, 18	3imtr4i 257	. 2 |- (a e. Nn -> (a +o 1c) e. Nn )
20	2, 19	vtoclga 2921	1 |- (A e. Nn -> (A +o 1c) e. Nn )

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2615  |^|cint 3927  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +o cplc 4376

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 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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-pr 3743  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-p6 4192  df-sik 4193  df-ssetk 4194  df-addc 4379  df-nnc 4380

This theorem is referenced by:  1cnnc  4409  peano5  4410  nnc0suc  4413  nncaddccl  4420  ltfinirr  4458  ltfintr  4460  lefinlteq  4464  ltfintri  4467  ltlefin  4469  ssfin  4471  ncfinraise  4482  ncfinlower  4484  tfinsuc  4499  oddnn  4508  sucoddeven  4512  evenodddisj  4517  oddtfin  4519  sfindbl  4531  sfintfin  4533  peano4  4558  phi11lem1  4596  2nnc  6168  nclenn  6250  nnltp1c  6263  nmembers1lem3  6271  nncdiv3  6278  nnc3n3p1  6279  nnc3n3p2  6280  nnc3p1n3p2  6281  nchoicelem1  6290  nchoicelem2  6291  nchoicelem12  6301  nchoicelem17  6306  frecxp  6315  frecsuc  6323