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| 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 12881 | . . 3 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) = {𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘}) | |
| 2 | 1 | eleq2d 2826 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑁 ∈ {𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘})) | 
| 3 | breq2 5146 | . . 3 ⊢ (𝑘 = 𝑁 → (𝑀 ≤ 𝑘 ↔ 𝑀 ≤ 𝑁)) | |
| 4 | 3 | elrab 3691 | . 2 ⊢ (𝑁 ∈ {𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘} ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | 
| 5 | 2, 4 | bitrdi 287 | 1 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2107 {crab 3435 class class class wbr 5142 ‘cfv 6560 ≤ cle 11297 ℤcz 12615 ℤ≥cuz 12879 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-cnex 11212 ax-resscn 11213 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-neg 11496 df-z 12616 df-uz 12880 | 
| This theorem is referenced by: eluz2 12885 eluz1i 12887 eluz 12893 uzid 12894 uzss 12902 eluzp1m1 12905 raluz 12939 rexuz 12941 preduz 13691 fi1uzind 14547 algcvga 16617 uzssico 32787 nndiffz1 32789 fzspl 32792 cycpmco2lem6 33152 cycpmconjslem2 33176 breprexplemc 34648 logblebd 41978 aks6d1c1 42118 aks6d1c2lem4 42129 lzunuz 42784 | 
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