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Mirrors > Home > ILE Home > Th. List > elfzel1 | GIF version |
Description: Membership in a finite set of sequential integer implies the lower bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
elfzel1 | ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuz 10020 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) | |
2 | eluzel2 9532 | . 2 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ‘cfv 5216 (class class class)co 5874 ℤcz 9252 ℤ≥cuz 9527 ...cfz 10007 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-neg 8130 df-z 9253 df-uz 9528 df-fz 10008 |
This theorem is referenced by: fzdisj 10051 fzrev2i 10085 fzrev3 10086 uznfz 10102 elfzmlbm 10130 fzoval 10147 iseqf1olemqcl 10485 iseqf1olemab 10488 iseqf1olemqf1o 10492 iseqf1olemqk 10493 iseqf1olemjpcl 10494 iseqf1olemqpcl 10495 iseqf1olemfvp 10496 seq3f1olemqsumkj 10497 seq3f1olemqsumk 10498 seq3f1olemqsum 10499 seq3f1olemstep 10500 bcp1nk 10741 |
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