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Mirrors > Home > MPE Home > Th. List > eluz1 | Structured version Visualization version GIF version |
Description: Membership in the upper set of integers starting at 𝑀. (Contributed by NM, 5-Sep-2005.) |
Ref | Expression |
---|---|
eluz1 | ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzval 12233 | . . 3 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) = {𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘}) | |
2 | 1 | eleq2d 2895 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑁 ∈ {𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘})) |
3 | breq2 5061 | . . 3 ⊢ (𝑘 = 𝑁 → (𝑀 ≤ 𝑘 ↔ 𝑀 ≤ 𝑁)) | |
4 | 3 | elrab 3677 | . 2 ⊢ (𝑁 ∈ {𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘} ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) |
5 | 2, 4 | syl6bb 288 | 1 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 {crab 3139 class class class wbr 5057 ‘cfv 6348 ≤ cle 10664 ℤcz 11969 ℤ≥cuz 12231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-cnex 10581 ax-resscn 10582 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-neg 10861 df-z 11970 df-uz 12232 |
This theorem is referenced by: eluz2 12237 eluz1i 12239 eluz 12245 uzid 12246 uzss 12253 eluzp1m1 12256 raluz 12284 rexuz 12286 preduz 13017 fi1uzind 13843 algcvga 15911 uzssico 30433 nndiffz1 30435 fzspl 30439 cycpmco2lem6 30700 cycpmconjslem2 30724 breprexplemc 31802 lzunuz 39243 |
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