Theorem peano1 4402

Description: Cardinal zero is a finite cardinal. Theorem X.1.4 of [Rosser] p. 276. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
peano1	⊢ 0c ∈ Nn

Proof of Theorem peano1

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

Step	Hyp	Ref	Expression
1		df-nnc 4379	. . . 4 ⊢ Nn = ∩{x ∣ (0c ∈ x ∧ ∀y ∈ x (y +c 1c) ∈ x)}
2	1	eleq2i 2417	. . 3 ⊢ (0c ∈ Nn ↔ 0c ∈ ∩{x ∣ (0c ∈ x ∧ ∀y ∈ x (y +c 1c) ∈ x)})
3		0cex 4392	. . . 4 ⊢ 0c ∈ V
4	3	elintab 3937	. . 3 ⊢ (0c ∈ ∩{x ∣ (0c ∈ x ∧ ∀y ∈ x (y +c 1c) ∈ x)} ↔ ∀x((0c ∈ x ∧ ∀y ∈ x (y +c 1c) ∈ x) → 0c ∈ x))
5	2, 4	bitri 240	. 2 ⊢ (0c ∈ Nn ↔ ∀x((0c ∈ x ∧ ∀y ∈ x (y +c 1c) ∈ x) → 0c ∈ x))
6		simpl 443	. 2 ⊢ ((0c ∈ x ∧ ∀y ∈ x (y +c 1c) ∈ x) → 0c ∈ x)
7	5, 6	mpgbir 1550	1 ⊢ 0c ∈ Nn

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∩cint 3926  1_cc1c 4134   Nn cnnc 4373  0_cc0c 4374   +_c cplc 4375

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-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-int 3927  df-0c 4377  df-nnc 4379

This theorem is referenced by:  1cnnc  4408  peano5  4409  nnc0suc  4412  0fin  4423  ltfinirr  4457  0cminle  4461  ltfinp1  4462  lefinlteq  4463  lefinrflx  4467  ncfinraise  4481  ncfinlower  4483  tfin0c  4497  tfin1c  4499  0ceven  4505  sfin01  4528  vfin1cltv  4547  0cnelphi  4597  ncssfin  6151  nclenn  6249  nncdiv3  6277  nnc3n3p1  6278  nchoicelem17  6305  frecxp  6314  frec0  6321