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Mirrors > Home > ILE Home > Th. List > uzid | GIF version |
Description: Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
Ref | Expression |
---|---|
uzid | ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 8249 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
2 | 1 | leidd 7506 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ≤ 𝑀) |
3 | 2 | ancli 306 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑀 ∈ ℤ ∧ 𝑀 ≤ 𝑀)) |
4 | eluz1 8477 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑀 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑀 ≤ 𝑀))) | |
5 | 3, 4 | mpbird 156 | 1 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ 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-in1 544 ax-in2 545 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-13 1404 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-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-pre-ltirr 6996 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 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-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 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-iota 4867 df-fun 4904 df-fv 4910 df-ov 5515 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-neg 7185 df-z 8246 df-uz 8474 |
This theorem is referenced by: uzn0 8488 uz11 8495 eluzfz1 8895 eluzfz2 8896 elfz3 8898 elfz1end 8919 fzssp1 8930 fzpred 8932 fzp1ss 8935 fzpr 8939 fztp 8940 elfz0add 8979 fzolb 9009 zpnn0elfzo 9063 fzosplitsnm1 9065 fzofzp1 9083 fzosplitsn 9089 fzostep1 9093 frec2uzuzd 9188 frecuzrdgrrn 9194 frec2uzrdg 9195 frecuzrdgrom 9196 iseqfn 9221 iseq1 9222 iseqcl 9223 iseqp1 9225 iseqfveq 9230 iseq1p 9239 iseqcaopr3 9240 iseqhomo 9248 rexuz3 9588 r19.2uz 9591 resqrexlemcvg 9617 resqrexlemgt0 9618 resqrexlemoverl 9619 cau3lem 9710 caubnd2 9713 climconst 9811 climuni 9814 climcau 9866 serif0 9871 ialgr0 9883 |
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