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Mirrors > Home > ILE Home > Th. List > eluzfz1 | GIF version |
Description: Membership in a finite set of sequential integers - special case. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
eluzfz1 | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 9181 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
2 | uzid 9190 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (ℤ≥‘𝑀)) |
4 | eluzfz 9642 | . 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ (𝑀...𝑁)) | |
5 | 3, 4 | mpancom 416 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1448 ‘cfv 5059 (class class class)co 5706 ℤcz 8906 ℤ≥cuz 9176 ...cfz 9631 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-pre-ltirr 7607 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-neg 7807 df-z 8907 df-uz 9177 df-fz 9632 |
This theorem is referenced by: elfz3 9655 fzm 9659 fzopth 9682 fz01or 9732 exfzdc 9858 seq3clss 10081 seq3fveq 10085 seq3shft2 10087 monoord 10090 monoord2 10091 iseqf1olemqk 10108 seq3f1olemqsumkj 10112 seq3f1olemp 10116 seq3id3 10121 ser3ge0 10131 seq3coll 10426 fsum1p 11026 telfsumo 11074 telfsumo2 11075 fsumparts 11078 mertenslem2 11144 phicl2 11682 inffz 12822 |
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