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Mirrors > Home > ILE Home > Th. List > elfzelzd | GIF version |
Description: A member of a finite set of sequential integers is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
elfzelzd.1 | ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) |
Ref | Expression |
---|---|
elfzelzd | ⊢ (𝜑 → 𝐾 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzelzd.1 | . 2 ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) | |
2 | elfzelz 10091 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2164 (class class class)co 5918 ℤcz 9317 ...cfz 10074 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-fv 5262 df-ov 5921 df-oprab 5922 df-mpo 5923 df-neg 8193 df-z 9318 df-uz 9593 df-fz 10075 |
This theorem is referenced by: seqf1oglem1 10590 seqf1oglem2 10591 seqfeq4g 10602 4sqlem12 12540 gausslemma2dlem1cl 15175 gausslemma2dlem1f1o 15176 gausslemma2dlem2 15178 gausslemma2dlem4 15180 lgsquadlem1 15191 |
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