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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 12822 | . 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 205 ∧ wa 396 ∈ wcel 2106 class class class wbr 5147 ‘cfv 6540 ≤ cle 11245 ℤcz 12554 ℤ≥cuz 12818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-cnex 11162 ax-resscn 11163 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7408 df-neg 11443 df-z 12555 df-uz 12819 |
This theorem is referenced by: eluzaddiOLD 12850 eluzsubiOLD 12852 eluz2b1 12899 fz0to4untppr 13600 faclbnd4lem1 14249 climcndslem1 15791 ef01bndlem 16123 sin01bnd 16124 cos01bnd 16125 sin01gt0 16129 dvradcnv 25924 bposlem3 26778 bposlem4 26779 bposlem5 26780 bposlem9 26784 istrkg3ld 27701 axlowdimlem16 28204 ballotlem2 33475 nn0prpwlem 35195 jm2.20nn 41721 stoweidlem17 44719 wallispilem4 44770 nn0o1gt2ALTV 46348 ackval42 47335 |
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