Theorem peano1 4403

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 4380	. . . 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 4393	. . . 4 ⊢ 0c ∈ V
4	3	elintab 3938	. . 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 2615  ∩cint 3927  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +_c 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-sn 4088

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 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-int 3928  df-0c 4378  df-nnc 4380

This theorem is referenced by:  1cnnc  4409  peano5  4410  nnc0suc  4413  0fin  4424  ltfinirr  4458  0cminle  4462  ltfinp1  4463  lefinlteq  4464  lefinrflx  4468  ncfinraise  4482  ncfinlower  4484  tfin0c  4498  tfin1c  4500  0ceven  4506  sfin01  4529  vfin1cltv  4548  0cnelphi  4598  ncssfin  6152  nclenn  6250  nncdiv3  6278  nnc3n3p1  6279  nchoicelem17  6306  frecxp  6315  frec0  6322