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Mirrors > Home > ILE Home > Th. List > elfzoel2 | GIF version |
Description: Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
Ref | Expression |
---|---|
elfzoel2 | ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fzo 9920 | . 2 ⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) | |
2 | 1 | elmpocl2 5970 | 1 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 (class class class)co 5774 1c1 7621 − cmin 7933 ℤcz 9054 ...cfz 9790 ..^cfzo 9919 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-fzo 9920 |
This theorem is referenced by: elfzoelz 9924 elfzo2 9927 elfzole1 9932 elfzolt2 9933 elfzolt3 9934 elfzolt2b 9935 elfzolt3b 9936 fzonel 9937 elfzouz2 9938 fzonnsub 9946 fzoss1 9948 fzospliti 9953 fzodisj 9955 fzoaddel 9969 fzosubel 9971 fzoend 9999 ssfzo12 10001 fzofzp1 10004 peano2fzor 10009 fzostep1 10014 iseqf1olemqk 10267 fzomaxdiflem 10884 fzo0dvdseq 11555 fzocongeq 11556 addmodlteqALT 11557 |
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