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Mirrors > Home > ILE Home > Th. List > eluz2 | GIF version |
Description: Membership in an upper set of integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show 𝑀 ∈ ℤ. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
eluz2 | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 9324 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
2 | simp1 981 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℤ) | |
3 | eluz1 9323 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) | |
4 | ibar 299 | . . . 4 ⊢ (𝑀 ∈ ℤ → ((𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) ↔ (𝑀 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)))) | |
5 | 3, 4 | bitrd 187 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)))) |
6 | 3anass 966 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) ↔ (𝑀 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) | |
7 | 5, 6 | syl6bbr 197 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) |
8 | 1, 2, 7 | pm5.21nii 693 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∧ w3a 962 ∈ wcel 1480 class class class wbr 3924 ‘cfv 5118 ≤ cle 7794 ℤcz 9047 ℤ≥cuz 9319 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-cnex 7704 ax-resscn 7705 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-neg 7929 df-z 9048 df-uz 9320 |
This theorem is referenced by: eluzuzle 9327 eluzelz 9328 eluzle 9331 uztrn 9335 eluzp1p1 9344 uznn0sub 9350 uz3m2nn 9361 1eluzge0 9362 2eluzge1 9364 raluz2 9367 rexuz2 9369 peano2uz 9371 nn0pzuz 9375 uzind4 9376 nn0ge2m1nnALT 9403 elfzuzb 9793 uzsubsubfz 9820 ige2m1fz 9883 4fvwrd4 9910 elfzo2 9920 elfzouz2 9931 fzossrbm1 9943 fzossfzop1 9982 ssfzo12bi 9995 elfzonelfzo 10000 elfzomelpfzo 10001 fzosplitprm1 10004 fzostep1 10007 fzind2 10009 flqword2 10055 fldiv4p1lem1div2 10071 uzennn 10202 seq3split 10245 iseqf1olemqk 10260 seq3f1olemqsumkj 10264 seq3f1olemqsumk 10265 seq3f1olemqsum 10266 bcval5 10502 seq3coll 10578 seq3shft 10603 resqrexlemoverl 10786 resqrexlemga 10788 fsum3cvg3 11158 fisumrev2 11208 isumshft 11252 cvgratnnlemseq 11288 cvgratnnlemabsle 11289 cvgratnnlemsumlt 11290 cvgratz 11294 oddge22np1 11567 nn0o 11593 dvdsnprmd 11795 prmgt1 11801 oddprmgt2 11803 oddprmge3 11804 strleund 12036 strleun 12037 |
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