Theorem uzn0 9771

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 9757	. . 3 |- ZZ>= : ZZ --> ~P ZZ
2		ffn 5482	. . 3 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
3		fvelrnb 5693	. . 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 9769	. . . . 5 |- ( k e. ZZ -> k e. ( ZZ>= ` k ) )
6		ne0i 3501	. . . . 5 |- ( k e. ( ZZ>= ` k ) -> ( ZZ>= ` k ) =/= (/) )
7	5, 6	syl 14	. . . 4 |- ( k e. ZZ -> ( ZZ>= ` k ) =/= (/) )
8		neeq1 2415	. . . 4 |- ( ( ZZ>= ` k ) = M -> ( ( ZZ>= ` k ) =/= (/) <-> M =/= (/) ) )
9	7, 8	syl5ibcom 155	. . 3 |- ( k e. ZZ -> ( ( ZZ>= ` k ) = M -> M =/= (/) ) )
10	9	rexlimiv 2644	. 2 |- ( E. k e. ZZ ( ZZ>= ` k ) = M -> M =/= (/) )
11	4, 10	sylbi 121	1 |- ( M e. ran ZZ>= -> M =/= (/) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   e. wcel 2202   =/= wne 2402   E.wrex 2511   (/)c0 3494   ~Pcpw 3652   ran crn 4726   Fn wfn 5321   -->wf 5322   ` cfv 5326   ZZcz 9478   ZZ>=cuz 9754

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-in1 619  ax-in2 620  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-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-pre-ltirr 8143

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-neg 8352  df-z 9479  df-uz 9755

This theorem is referenced by: (None)