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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 10099 | . 2 ⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) | |
2 | 1 | elmpocl2 6049 | 1 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2141 (class class class)co 5853 1c1 7775 − cmin 8090 ℤcz 9212 ...cfz 9965 ..^cfzo 10098 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-fzo 10099 |
This theorem is referenced by: elfzoelz 10103 elfzo2 10106 elfzole1 10111 elfzolt2 10112 elfzolt3 10113 elfzolt2b 10114 elfzolt3b 10115 fzonel 10116 elfzouz2 10117 fzonnsub 10125 fzoss1 10127 fzospliti 10132 fzodisj 10134 fzoaddel 10148 fzosubel 10150 fzoend 10178 ssfzo12 10180 fzofzp1 10183 peano2fzor 10188 fzostep1 10193 iseqf1olemqk 10450 fzomaxdiflem 11076 fzo0dvdseq 11817 fzocongeq 11818 addmodlteqALT 11819 |
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