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Mirrors > Home > ILE Home > Th. List > elfzoel1 | GIF version |
Description: Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
Ref | Expression |
---|---|
elfzoel1 | ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fzo 9913 | . 2 ⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) | |
2 | 1 | elmpocl1 5962 | 1 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 (class class class)co 5767 1c1 7614 − cmin 7926 ℤcz 9047 ...cfz 9783 ..^cfzo 9912 |
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 |
This theorem depends on definitions: df-bi 116 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-v 2683 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-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-fzo 9913 |
This theorem is referenced by: elfzoelz 9917 fzoval 9918 elfzo2 9920 elfzole1 9925 elfzolt2 9926 elfzolt3 9927 elfzolt3b 9929 fzospliti 9946 fzoaddel 9962 fzosubel 9964 fzosubel3 9966 fzofzp1 9997 fzostep1 10007 fzomaxdiflem 10877 fzocongeq 11545 |
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