Theorem uzn0 9833

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 9819	. . 3 |- ZZ>= : ZZ --> ~P ZZ
2		ffn 5489	. . 3 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
3		fvelrnb 5702	. . 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 9831	. . . . 5 |- ( k e. ZZ -> k e. ( ZZ>= ` k ) )
6		ne0i 3503	. . . . 5 |- ( k e. ( ZZ>= ` k ) -> ( ZZ>= ` k ) =/= (/) )
7	5, 6	syl 14	. . . 4 |- ( k e. ZZ -> ( ZZ>= ` k ) =/= (/) )
8		neeq1 2416	. . . 4 |- ( ( ZZ>= ` k ) = M -> ( ( ZZ>= ` k ) =/= (/) <-> M =/= (/) ) )
9	7, 8	syl5ibcom 155	. . 3 |- ( k e. ZZ -> ( ( ZZ>= ` k ) = M -> M =/= (/) ) )
10	9	rexlimiv 2645	. 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 1398   e. wcel 2202   =/= wne 2403   E.wrex 2512   (/)c0 3496   ~Pcpw 3656   ran crn 4732   Fn wfn 5328   -->wf 5329   ` cfv 5333   ZZcz 9540   ZZ>=cuz 9816

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 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-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-pre-ltirr 8204

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-neg 8412  df-z 9541  df-uz 9817

This theorem is referenced by: (None)