| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > uztrn2 | Structured version Visualization version 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 2833 | . . 3 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ≥‘𝐾)) |
| 3 | uztrn 12896 | . . . 4 ⊢ ((𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → 𝑀 ∈ (ℤ≥‘𝐾)) | |
| 4 | 3 | ancoms 458 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝐾) ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝐾)) |
| 5 | 2, 4 | sylanb 581 | . 2 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝐾)) |
| 6 | 5, 1 | eleqtrrdi 2852 | 1 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ 𝑍) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 ℤ≥cuz 12878 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-neg 11495 df-z 12614 df-uz 12879 |
| This theorem is referenced by: eluznn0 12959 eluznn 12960 elfzuz2 13569 rexuz3 15387 r19.29uz 15389 r19.2uz 15390 clim2 15540 clim2c 15541 clim0c 15543 rlimclim1 15581 2clim 15608 climabs0 15621 climcn1 15628 climcn2 15629 climsqz 15677 climsqz2 15678 clim2ser 15691 clim2ser2 15692 climub 15698 climsup 15706 caurcvg2 15714 serf0 15717 iseraltlem1 15718 iseralt 15721 cvgcmp 15852 cvgcmpce 15854 isumsup2 15882 mertenslem1 15920 clim2div 15925 ntrivcvgfvn0 15935 ntrivcvgmullem 15937 fprodeq0 16011 lmbrf 23268 lmss 23306 lmres 23308 txlm 23656 uzrest 23905 lmmcvg 25295 lmmbrf 25296 iscau4 25313 iscauf 25314 caucfil 25317 iscmet3lem3 25324 iscmet3lem1 25325 lmle 25335 lmclim 25337 mbflimsup 25701 ulm2 26428 ulmcaulem 26437 ulmcau 26438 ulmss 26440 ulmdvlem1 26443 ulmdvlem3 26445 mtest 26447 itgulm 26451 logfaclbnd 27266 bposlem6 27333 caures 37767 caushft 37768 dvgrat 44331 cvgdvgrat 44332 climinf 45621 clim2f 45651 clim2cf 45665 clim0cf 45669 clim2f2 45685 fnlimfvre 45689 allbutfifvre 45690 limsupvaluz2 45753 limsupreuzmpt 45754 supcnvlimsup 45755 climuzlem 45758 climisp 45761 climrescn 45763 climxrrelem 45764 climxrre 45765 limsupgtlem 45792 liminfreuzlem 45817 liminfltlem 45819 liminflimsupclim 45822 xlimpnfxnegmnf 45829 liminflbuz2 45830 liminfpnfuz 45831 liminflimsupxrre 45832 xlimmnfvlem2 45848 xlimmnfv 45849 xlimpnfvlem2 45852 xlimpnfv 45853 xlimmnfmpt 45858 xlimpnfmpt 45859 climxlim2lem 45860 xlimpnfxnegmnf2 45873 meaiuninc3v 46499 smflimlem1 46786 smflimlem2 46787 smflimlem3 46788 smflimmpt 46825 smflimsuplem4 46838 smflimsuplem7 46841 smflimsupmpt 46844 smfliminfmpt 46847 |
| Copyright terms: Public domain | W3C validator |