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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 10051 | . 2 ⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) | |
2 | 1 | elmpocl2 6022 | 1 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2128 (class class class)co 5826 1c1 7735 − cmin 8050 ℤcz 9172 ...cfz 9918 ..^cfzo 10050 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-ov 5829 df-oprab 5830 df-mpo 5831 df-fzo 10051 |
This theorem is referenced by: elfzoelz 10055 elfzo2 10058 elfzole1 10063 elfzolt2 10064 elfzolt3 10065 elfzolt2b 10066 elfzolt3b 10067 fzonel 10068 elfzouz2 10069 fzonnsub 10077 fzoss1 10079 fzospliti 10084 fzodisj 10086 fzoaddel 10100 fzosubel 10102 fzoend 10130 ssfzo12 10132 fzofzp1 10135 peano2fzor 10140 fzostep1 10145 iseqf1olemqk 10402 fzomaxdiflem 11023 fzo0dvdseq 11761 fzocongeq 11762 addmodlteqALT 11763 |
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