Theorem elz 9080

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 2147	. . 3 |- ( x = N -> ( x = 0 <-> N = 0 ) )
2		eleq1 2203	. . 3 |- ( x = N -> ( x e. NN <-> N e. NN ) )
3		negeq 7979	. . . 4 |- ( x = N -> -u x = -u N )
4	3	eleq1d 2209	. . 3 |- ( x = N -> ( -u x e. NN <-> -u N e. NN ) )
5	1, 2, 4	3orbi123d 1290	. 2 |- ( x = N -> ( ( x = 0 \/ x e. NN \/ -u x e. NN ) <-> ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
6		df-z 9079	. 2 |- ZZ = { x e. RR | ( x = 0 \/ x e. NN \/ -u x e. NN ) }
7	5, 6	elrab2 2847	1 |- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   \/ w3o 962   = wceq 1332   e. wcel 1481   RRcr 7643   0cc0 7644   -ucneg 7958   NNcn 8744   ZZcz 9078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-rab 2426  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139  df-ov 5785  df-neg 7960  df-z 9079

This theorem is referenced by:  nnnegz  9081  zre  9082  elnnz  9088  0z  9089  elnn0z  9091  elznn0nn  9092  elznn0  9093  elznn  9094  znegcl  9109  zaddcl  9118  ztri3or0  9120  zeo  9180  addmodlteq  10202