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Mirrors > Home > ILE Home > Th. List > eluz | GIF version |
Description: Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.) |
Ref | Expression |
---|---|
eluz | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz1 9461 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) | |
2 | 1 | baibd 913 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2135 class class class wbr 3976 ‘cfv 5182 ≤ cle 7925 ℤcz 9182 ℤ≥cuz 9457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-cnex 7835 ax-resscn 7836 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-ov 5839 df-neg 8063 df-z 9183 df-uz 9458 |
This theorem is referenced by: uzneg 9475 uztric 9478 uzm1 9487 eluzdc 9539 fzn 9967 fzsplit2 9975 fznn 10014 uzsplit 10017 elfz2nn0 10037 fzouzsplit 10104 exfzdc 10165 fzfig 10355 faclbnd 10643 seq3coll 10741 cvg1nlemcau 10912 cvg1nlemres 10913 summodclem2a 11308 fsum0diaglem 11367 mertenslemi1 11462 prodmodclem2a 11503 zsupcllemstep 11863 zsupcl 11865 infssuzex 11867 pcpremul 12204 pcdvdsb 12230 pcadd 12250 pcfac 12259 pcbc 12260 uzdcinzz 13520 |
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