Theorem elz 9646

Description: Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)

Assertion

Ref	Expression
elz	|- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )

Proof of Theorem elz

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2245	. . 3 |- ( x = N -> ( x = 0 <-> N = 0 ) )
2		eleq1 2301	. . 3 |- ( x = N -> ( x e. NN <-> N e. NN ) )
3		negeq 8519	. . . 4 |- ( x = N -> -u x = -u N )
4	3	eleq1d 2307	. . 3 |- ( x = N -> ( -u x e. NN <-> -u N e. NN ) )
5	1, 2, 4	3orbi123d 1352	. 2 |- ( x = N -> ( ( x = 0 \/ x e. NN \/ -u x e. NN ) <-> ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
6		df-z 9645	. 2 |- ZZ = { x e. RR | ( x = 0 \/ x e. NN \/ -u x e. NN ) }
7	5, 6	elrab2 2985	1 |- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   \/ w3o 1008   = wceq 1402   e. wcel 2209   RRcr 8178   0cc0 8179   -ucneg 8498   NNcn 9304   ZZcz 9644

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088  df-neg 8500  df-z 9645

This theorem is used by:  nnnegz  9647  zre  9648  elnnz  9654  0z  9655  elnn0z  9657  elznn0nn  9658  elznn0  9659  elznn  9660  znegcl  9675  zaddcl  9684  ztri3or0  9686  zeo  9751  addmodlteq  10835  zabsle1  16118