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Mirrors > Home > MPE Home > Th. List > eluzfz2b | Structured version Visualization version GIF version |
Description: Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.) |
Ref | Expression |
---|---|
eluzfz2b | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑁 ∈ (𝑀...𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzfz2 13263 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) | |
2 | elfzuz 13251 | . 2 ⊢ (𝑁 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) | |
3 | 1, 2 | impbii 208 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑁 ∈ (𝑀...𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2110 ‘cfv 6432 (class class class)co 7271 ℤ≥cuz 12581 ...cfz 13238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-pre-lttri 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-nel 3052 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-oprab 7275 df-mpo 7276 df-1st 7824 df-2nd 7825 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-neg 11208 df-z 12320 df-uz 12582 df-fz 13239 |
This theorem is referenced by: smupval 16193 smueqlem 16195 smumul 16198 efgtlen 19330 dvntaylp 25528 taylthlem1 25530 |
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