Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > eluzelz | GIF version |
Description: A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.) |
Ref | Expression |
---|---|
eluzelz | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2 8479 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
2 | 1 | simp2bi 920 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 class class class wbr 3764 ‘cfv 4902 ≤ cle 7061 ℤcz 8245 ℤ≥cuz 8473 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-cnex 6975 ax-resscn 6976 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 df-ov 5515 df-neg 7185 df-z 8246 df-uz 8474 |
This theorem is referenced by: eluzelre 8483 uztrn 8489 uzneg 8491 uzssz 8492 uzss 8493 eluzp1l 8497 eluzaddi 8499 eluzsubi 8500 eluzadd 8501 eluzsub 8502 uzm1 8503 uzin 8505 uzind4 8531 uz2mulcl 8545 elfz5 8882 elfzel2 8888 elfzelz 8890 eluzfz2 8896 peano2fzr 8901 fzsplit2 8914 fzopth 8924 fzsuc 8931 elfzp1 8934 fzdifsuc 8943 uzsplit 8954 uzdisj 8955 fzm1 8962 fzneuz 8963 uznfz 8965 nn0disj 8995 elfzo3 9019 fzoss2 9028 fzouzsplit 9035 eluzgtdifelfzo 9053 fzosplitsnm1 9065 fzofzp1b 9084 elfzonelfzo 9086 fzosplitsn 9089 fzisfzounsn 9092 frec2uzltd 9189 frecfzen2 9204 iseqfveq2 9228 iseqfeq2 9229 iseqshft2 9232 monoord 9235 monoord2 9236 isermono 9237 iseqsplit 9238 iseqid 9247 leexp2a 9307 expnlbnd2 9374 rexuz3 9588 r19.2uz 9591 cau4 9712 caubnd2 9713 clim 9802 climshft2 9827 climaddc1 9849 climmulc2 9851 climsubc1 9852 climsubc2 9853 clim2iser 9857 clim2iser2 9858 iiserex 9859 climlec2 9861 climub 9864 climcau 9866 climcaucn 9870 serif0 9871 |
Copyright terms: Public domain | W3C validator |