Theorem uzn0 9341

Description: The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)

Assertion

Ref	Expression
uzn0	|- ( M e. ran ZZ>= -> M =/= (/) )

Proof of Theorem uzn0

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uzf 9329	. . 3 |- ZZ>= : ZZ --> ~P ZZ
2		ffn 5272	. . 3 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
3		fvelrnb 5469	. . 3 |- ( ZZ>= Fn ZZ -> ( M e. ran ZZ>= <-> E. k e. ZZ ( ZZ>= ` k ) = M ) )
4	1, 2, 3	mp2b 8	. 2 |- ( M e. ran ZZ>= <-> E. k e. ZZ ( ZZ>= ` k ) = M )
5		uzid 9340	. . . . 5 |- ( k e. ZZ -> k e. ( ZZ>= ` k ) )
6		ne0i 3369	. . . . 5 |- ( k e. ( ZZ>= ` k ) -> ( ZZ>= ` k ) =/= (/) )
7	5, 6	syl 14	. . . 4 |- ( k e. ZZ -> ( ZZ>= ` k ) =/= (/) )
8		neeq1 2321	. . . 4 |- ( ( ZZ>= ` k ) = M -> ( ( ZZ>= ` k ) =/= (/) <-> M =/= (/) ) )
9	7, 8	syl5ibcom 154	. . 3 |- ( k e. ZZ -> ( ( ZZ>= ` k ) = M -> M =/= (/) ) )
10	9	rexlimiv 2543	. 2 |- ( E. k e. ZZ ( ZZ>= ` k ) = M -> M =/= (/) )
11	4, 10	sylbi 120	1 |- ( M e. ran ZZ>= -> M =/= (/) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   =/= wne 2308   E.wrex 2417   (/)c0 3363   ~Pcpw 3510   ran crn 4540   Fn wfn 5118   -->wf 5119   ` cfv 5123   ZZcz 9054   ZZ>=cuz 9326

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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-pre-ltirr 7732

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-neg 7936  df-z 9055  df-uz 9327

This theorem is referenced by: (None)