Theorem elz 9480

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 2238	. . 3 |- ( x = N -> ( x = 0 <-> N = 0 ) )
2		eleq1 2294	. . 3 |- ( x = N -> ( x e. NN <-> N e. NN ) )
3		negeq 8371	. . . 4 |- ( x = N -> -u x = -u N )
4	3	eleq1d 2300	. . 3 |- ( x = N -> ( -u x e. NN <-> -u N e. NN ) )
5	1, 2, 4	3orbi123d 1347	. 2 |- ( x = N -> ( ( x = 0 \/ x e. NN \/ -u x e. NN ) <-> ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
6		df-z 9479	. 2 |- ZZ = { x e. RR | ( x = 0 \/ x e. NN \/ -u x e. NN ) }
7	5, 6	elrab2 2965	1 |- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   \/ w3o 1003   = wceq 1397   e. wcel 2202   RRcr 8030   0cc0 8031   -ucneg 8350   NNcn 9142   ZZcz 9478

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6020  df-neg 8352  df-z 9479

This theorem is referenced by:  nnnegz  9481  zre  9482  elnnz  9488  0z  9489  elnn0z  9491  elznn0nn  9492  elznn0  9493  elznn  9494  znegcl  9509  zaddcl  9518  ztri3or0  9520  zeo  9584  addmodlteq  10659  zabsle1  15727