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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 11567 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) | |
2 | 1 | baibd 946 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 ≤ cle 9954 ℤcz 11254 ℤ≥cuz 11563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-neg 10148 df-z 11255 df-uz 11564 |
This theorem is referenced by: uzneg 11582 uztric 11585 uzwo3 11659 fzn 12228 fzsplit2 12237 fznn 12278 uzsplit 12281 elfz2nn0 12300 fzouzsplit 12372 faclbnd 12939 bcval5 12967 fz1isolem 13102 seqcoll 13105 rexuzre 13940 caurcvg 14255 caucvg 14257 summolem2a 14293 fsum0diaglem 14350 climcnds 14422 mertenslem1 14455 ntrivcvgmullem 14472 prodmolem2a 14503 ruclem10 14807 eulerthlem2 15325 pcpremul 15386 pcdvdsb 15411 pcadd 15431 pcfac 15441 pcbc 15442 prmunb 15456 prmreclem5 15462 vdwnnlem3 15539 lt6abl 18119 ovolunlem1a 23071 mbflimsup 23239 plyco0 23752 plyeq0lem 23770 aannenlem1 23887 aaliou3lem2 23902 aaliou3lem8 23904 chtublem 24736 bcmax 24803 bpos1lem 24807 bposlem1 24809 axlowdimlem16 25637 extwwlkfablem2 26605 fzsplit3 28940 ballotlem2 29877 ballotlemimin 29894 elfzm12 30823 poimirlem3 32582 poimirlem4 32583 poimirlem28 32607 mblfinlem2 32617 incsequz 32714 incsequz2 32715 nacsfix 36293 ellz1 36348 eluzrabdioph 36388 monotuz 36524 expdiophlem1 36606 nznngen 37537 fzisoeu 38455 fmul01 38647 climsuselem1 38674 climsuse 38675 iblspltprt 38865 itgspltprt 38871 wallispilem5 38962 stirlinglem8 38974 dirkertrigeqlem1 38991 fourierdlem12 39012 ssfz12 40351 |
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