Theorem elz 9255

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 2184	. . 3 |- ( x = N -> ( x = 0 <-> N = 0 ) )
2		eleq1 2240	. . 3 |- ( x = N -> ( x e. NN <-> N e. NN ) )
3		negeq 8150	. . . 4 |- ( x = N -> -u x = -u N )
4	3	eleq1d 2246	. . 3 |- ( x = N -> ( -u x e. NN <-> -u N e. NN ) )
5	1, 2, 4	3orbi123d 1311	. 2 |- ( x = N -> ( ( x = 0 \/ x e. NN \/ -u x e. NN ) <-> ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
6		df-z 9254	. 2 |- ZZ = { x e. RR | ( x = 0 \/ x e. NN \/ -u x e. NN ) }
7	5, 6	elrab2 2897	1 |- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   \/ w3o 977   = wceq 1353   e. wcel 2148   RRcr 7810   0cc0 7811   -ucneg 8129   NNcn 8919   ZZcz 9253

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-rab 2464  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-iota 5179  df-fv 5225  df-ov 5878  df-neg 8131  df-z 9254

This theorem is referenced by:  nnnegz  9256  zre  9257  elnnz  9263  0z  9264  elnn0z  9266  elznn0nn  9267  elznn0  9268  elznn  9269  znegcl  9284  zaddcl  9293  ztri3or0  9295  zeo  9358  addmodlteq  10398  zabsle1  14403