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Mirrors > Home > MPE Home > Th. List > eluz | Structured version Visualization version 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 12239 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) | |
2 | 1 | baibd 542 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2108 class class class wbr 5057 ‘cfv 6348 ≤ cle 10668 ℤcz 11973 ℤ≥cuz 12235 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-cnex 10585 ax-resscn 10586 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 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 7151 df-neg 10865 df-z 11974 df-uz 12236 |
This theorem is referenced by: uzneg 12255 uztric 12258 uzwo3 12335 fzn 12915 fzsplit2 12924 fznn 12967 uzsplit 12971 elfz2nn0 12990 fzouzsplit 13064 faclbnd 13642 bcval5 13670 fz1isolem 13811 seqcoll 13814 rexuzre 14704 caurcvg 15025 caucvg 15027 summolem2a 15064 fsum0diaglem 15123 climcnds 15198 mertenslem1 15232 ntrivcvgmullem 15249 prodmolem2a 15280 ruclem10 15584 eulerthlem2 16111 pcpremul 16172 pcdvdsb 16197 pcadd 16217 pcfac 16227 pcbc 16228 prmunb 16242 prmreclem5 16248 vdwnnlem3 16325 lt6abl 19007 ovolunlem1a 24089 mbflimsup 24259 plyco0 24774 plyeq0lem 24792 aannenlem1 24909 aaliou3lem2 24924 aaliou3lem8 24926 chtublem 25779 bcmax 25846 bpos1lem 25850 bposlem1 25852 axlowdimlem16 26735 fzsplit3 30509 cycpmco2lem7 30767 ballotlem2 31739 ballotlemimin 31756 breprexplemc 31896 elfzm12 32911 poimirlem3 34887 poimirlem4 34888 poimirlem28 34912 mblfinlem2 34922 incsequz 35015 incsequz2 35016 nacsfix 39300 ellz1 39355 eluzrabdioph 39394 monotuz 39529 expdiophlem1 39609 nznngen 40639 fzisoeu 41557 fmul01 41851 climsuselem1 41878 climsuse 41879 iblspltprt 42248 itgspltprt 42254 wallispilem5 42345 stirlinglem8 42357 dirkertrigeqlem1 42374 fourierdlem12 42395 ssfz12 43505 |
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