Theorem elz 9319

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 2200	. . 3 |- ( x = N -> ( x = 0 <-> N = 0 ) )
2		eleq1 2256	. . 3 |- ( x = N -> ( x e. NN <-> N e. NN ) )
3		negeq 8212	. . . 4 |- ( x = N -> -u x = -u N )
4	3	eleq1d 2262	. . 3 |- ( x = N -> ( -u x e. NN <-> -u N e. NN ) )
5	1, 2, 4	3orbi123d 1322	. 2 |- ( x = N -> ( ( x = 0 \/ x e. NN \/ -u x e. NN ) <-> ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
6		df-z 9318	. 2 |- ZZ = { x e. RR | ( x = 0 \/ x e. NN \/ -u x e. NN ) }
7	5, 6	elrab2 2919	1 |- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   \/ w3o 979   = wceq 1364   e. wcel 2164   RRcr 7871   0cc0 7872   -ucneg 8191   NNcn 8982   ZZcz 9317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921  df-neg 8193  df-z 9318

This theorem is referenced by:  nnnegz  9320  zre  9321  elnnz  9327  0z  9328  elnn0z  9330  elznn0nn  9331  elznn0  9332  elznn  9333  znegcl  9348  zaddcl  9357  ztri3or0  9359  zeo  9422  addmodlteq  10469  zabsle1  15115