Theorem uzn0 9334

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 9322	. . 3 |- ZZ>= : ZZ --> ~P ZZ
2		ffn 5267	. . 3 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
3		fvelrnb 5462	. . 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 9333	. . . . 5 |- ( k e. ZZ -> k e. ( ZZ>= ` k ) )
6		ne0i 3364	. . . . 5 |- ( k e. ( ZZ>= ` k ) -> ( ZZ>= ` k ) =/= (/) )
7	5, 6	syl 14	. . . 4 |- ( k e. ZZ -> ( ZZ>= ` k ) =/= (/) )
8		neeq1 2319	. . . 4 |- ( ( ZZ>= ` k ) = M -> ( ( ZZ>= ` k ) =/= (/) <-> M =/= (/) ) )
9	7, 8	syl5ibcom 154	. . 3 |- ( k e. ZZ -> ( ( ZZ>= ` k ) = M -> M =/= (/) ) )
10	9	rexlimiv 2541	. 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 2306   E.wrex 2415   (/)c0 3358   ~Pcpw 3505   ran crn 4535   Fn wfn 5113   -->wf 5114   ` cfv 5118   ZZcz 9047   ZZ>=cuz 9319

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 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-pre-ltirr 7725

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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-ov 5770  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-neg 7929  df-z 9048  df-uz 9320

This theorem is referenced by: (None)