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Mirrors > Home > MPE Home > Th. List > Mathboxes > elfzelzd | Structured version Visualization version GIF version |
Description: A member of a finite set of sequential integer 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 12909 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 (class class class)co 7156 ℤcz 11982 ...cfz 12893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-neg 10873 df-z 11983 df-uz 12245 df-fz 12894 |
This theorem is referenced by: etransclem1 42540 etransclem3 42542 etransclem7 42546 etransclem10 42549 etransclem15 42554 etransclem21 42560 etransclem22 42561 etransclem24 42563 etransclem25 42564 etransclem35 42574 etransclem37 42576 etransclem38 42577 |
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