Theorem eluzelz2 39627

Description: A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypothesis

Ref	Expression
eluzelz2.1	|- Z = ( ZZ>= ` M )

Assertion

Ref	Expression
eluzelz2	|- ( N e. Z -> N e. ZZ )

Proof of Theorem eluzelz2

Step	Hyp	Ref	Expression
1		eluzelz2.1	. . . 4 |- Z = ( ZZ>= ` M )
2	1	eleq2i 2693	. . 3 |- ( N e. Z <-> N e. ( ZZ>= ` M ) )
3	2	biimpi 206	. 2 |- ( N e. Z -> N e. ( ZZ>= ` M ) )
4		eluzelz 11697	. 2 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
5	3, 4	syl 17	1 |- ( N e. Z -> N e. ZZ )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888   ZZcz 11377   ZZ>=cuz 11687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-neg 10269  df-z 11378  df-uz 11688

This theorem is referenced by:  eluzelz2d  39640  uzublem  39657  uzinico  39787  limsupubuzlem  39944  limsupmnfuzlem  39958  limsupre3uzlem  39967  limsupvaluz2  39970  supcnvlimsup  39972  xlimclim2lem  40065  climxlim2  40072  smflimmpt  41016  smflimsuplem3  41028  smflimsuplem4  41029  smflimsuplem5  41030  smflimsuplem6  41031  smflimsuplem7  41032  smflimsuplem8  41033  smflimsupmpt  41035  smfliminflem  41036  smfliminfmpt  41038