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Mirrors > Home > MPE Home > Th. List > elfz4 | Structured version Visualization version GIF version |
Description: Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
elfz4 | ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝐾 ∈ (𝑀...𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfz2 13521 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) | |
2 | 1 | biimpri 227 | 1 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝐾 ∈ (𝑀...𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5141 (class class class)co 7414 ≤ cle 11277 ℤcz 12586 ...cfz 13514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5292 ax-nul 5299 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-fv 6549 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7989 df-2nd 7990 df-neg 11475 df-z 12587 df-fz 13515 |
This theorem is referenced by: 4bc2eq6 14318 isprm7 16676 axlowdimlem6 28774 |
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