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Mirrors > Home > ILE Home > Th. List > eluzle | GIF version |
Description: Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) |
Ref | Expression |
---|---|
eluzle | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2 8625 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
2 | 1 | simp3bi 955 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1433 class class class wbr 3785 ‘cfv 4922 ≤ cle 7154 ℤcz 8351 ℤ≥cuz 8619 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-cnex 7067 ax-resscn 7068 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-ov 5535 df-neg 7282 df-z 8352 df-uz 8620 |
This theorem is referenced by: uztrn 8635 uzneg 8637 uzss 8639 uz11 8641 eluzp1l 8643 uzm1 8649 uzin 8651 uzind4 8676 elfz5 9037 elfzle1 9046 elfzle2 9047 elfzle3 9049 uzsplit 9109 uzdisj 9110 uznfz 9120 elfz2nn0 9128 uzsubfz0 9140 nn0disj 9148 fzouzdisj 9189 elfzonelfzo 9239 mulp1mod1 9367 m1modge3gt1 9373 cvg1nlemcau 9870 resqrexlemcvg 9905 resqrexlemga 9909 infssuzex 10345 dvdsbnd 10348 ncoprmgcdne1b 10471 |
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