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| Mirrors > Home > MPE Home > Th. List > eluz1i | Structured version Visualization version GIF version | ||
| Description: Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.) |
| Ref | Expression |
|---|---|
| eluz.1 | ⊢ 𝑀 ∈ ℤ |
| Ref | Expression |
|---|---|
| eluz1i | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eluz.1 | . 2 ⊢ 𝑀 ∈ ℤ | |
| 2 | eluz1 12882 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 ≤ cle 11296 ℤcz 12613 ℤ≥cuz 12878 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-neg 11495 df-z 12614 df-uz 12879 |
| This theorem is referenced by: eluzaddiOLD 12910 eluzsubiOLD 12912 eluz2b1 12961 faclbnd4lem1 14332 climcndslem1 15885 ef01bndlem 16220 sin01bnd 16221 cos01bnd 16222 sin01gt0 16226 dvradcnv 26464 bposlem3 27330 bposlem4 27331 bposlem5 27332 bposlem9 27336 istrkg3ld 28469 axlowdimlem16 28972 2sqr3minply 33791 ballotlem2 34491 nn0prpwlem 36323 jm2.20nn 43009 stoweidlem17 46032 wallispilem4 46083 nn0o1gt2ALTV 47681 ackval42 48617 |
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