Theorem elnn0 7959

Description: Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)

Assertion

Ref	Expression
elnn0	⊢ (A ∈ ℕ0 ↔ (A ∈ ℕ ∨ A = 0))

Proof of Theorem elnn0

Step	Hyp	Ref	Expression
1		df-n0 7958	. . 3 ⊢ ℕ0 = (ℕ ∪ {0})
2	1	eleq2i 2101	. 2 ⊢ (A ∈ ℕ0 ↔ A ∈ (ℕ ∪ {0}))
3		elun 3078	. 2 ⊢ (A ∈ (ℕ ∪ {0}) ↔ (A ∈ ℕ ∨ A ∈ {0}))
4		c0ex 6819	. . . 4 ⊢ 0 ∈ V
5	4	elsnc2 3397	. . 3 ⊢ (A ∈ {0} ↔ A = 0)
6	5	orbi2i 678	. 2 ⊢ ((A ∈ ℕ ∨ A ∈ {0}) ↔ (A ∈ ℕ ∨ A = 0))
7	2, 3, 6	3bitri 195	1 ⊢ (A ∈ ℕ0 ↔ (A ∈ ℕ ∨ A = 0))

Syntax hints:   ↔ wb 98   ∨ wo 628   = wceq 1242   ∈ wcel 1390   ∪ cun 2909  {csn 3367  0cc0 6711  ℕcn 7695  ℕ₀cn0 7957

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-1cn 6776  ax-icn 6778  ax-addcl 6779  ax-mulcl 6781  ax-i2m1 6788

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-n0 7958

This theorem is referenced by:  0nn0  7972  nn0ge0  7983  nnnn0addcl  7988  nnm1nn0  7999  elnnnn0b  8002  elnn0z  8034  elznn0nn  8035  elznn0  8036  elznn  8037  nn0ind-raph  8131  expp1  8916  expnegap0  8917  expcllem  8920