Theorem elz 9542

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 8431	. . . 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 1348	. 2 |- ( x = N -> ( ( x = 0 \/ x e. NN \/ -u x e. NN ) <-> ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
6		df-z 9541	. 2 |- ZZ = { x e. RR | ( x = 0 \/ x e. NN \/ -u x e. NN ) }
7	5, 6	elrab2 2966	1 |- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   \/ w3o 1004   = wceq 1398   e. wcel 2202   RRcr 8091   0cc0 8092   -ucneg 8410   NNcn 9202   ZZcz 9540

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-rab 2520  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031  df-neg 8412  df-z 9541

This theorem is referenced by:  nnnegz  9543  zre  9544  elnnz  9550  0z  9551  elnn0z  9553  elznn0nn  9554  elznn0  9555  elznn  9556  znegcl  9571  zaddcl  9580  ztri3or0  9582  zeo  9646  addmodlteq  10723  zabsle1  15818