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Mirrors > Home > ILE Home > Th. List > uztrn2 | GIF version |
Description: Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.) |
Ref | Expression |
---|---|
uztrn2.1 | ⊢ 𝑍 = (ℤ≥‘𝐾) |
Ref | Expression |
---|---|
uztrn2 | ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uztrn2.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝐾) | |
2 | 1 | eleq2i 2244 | . . 3 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ≥‘𝐾)) |
3 | uztrn 9530 | . . . 4 ⊢ ((𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → 𝑀 ∈ (ℤ≥‘𝐾)) | |
4 | 3 | ancoms 268 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝐾) ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝐾)) |
5 | 2, 4 | sylanb 284 | . 2 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝐾)) |
6 | 5, 1 | eleqtrrdi 2271 | 1 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ 𝑍) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ‘cfv 5212 ℤ≥cuz 9514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7890 ax-resscn 7891 ax-pre-ltwlin 7912 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-fv 5220 df-ov 5872 df-pnf 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 df-neg 8118 df-z 9240 df-uz 9515 |
This theorem is referenced by: eluznn0 9585 eluznn 9586 elfzuz2 10012 rexuz3 10980 r19.29uz 10982 r19.2uz 10983 clim2 11272 clim2c 11273 clim0c 11275 2clim 11290 climabs0 11296 climcn1 11297 climcn2 11298 climsqz 11324 climsqz2 11325 clim2ser 11326 clim2ser2 11327 climub 11333 serf0 11341 mertenslemi1 11524 clim2divap 11529 fprodntrivap 11573 fprodeq0 11606 lmbrf 13375 lmss 13406 lmres 13408 txlm 13439 |
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