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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 9615 | . 2 ⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) | |
2 | 1 | elmpt2cl2 5858 | 1 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1439 (class class class)co 5666 1c1 7412 − cmin 7714 ℤcz 8811 ...cfz 9485 ..^cfzo 9614 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-fzo 9615 |
This theorem is referenced by: elfzoelz 9619 elfzo2 9622 elfzole1 9627 elfzolt2 9628 elfzolt3 9629 elfzolt2b 9630 elfzolt3b 9631 fzonel 9632 elfzouz2 9633 fzonnsub 9641 fzoss1 9643 fzospliti 9648 fzodisj 9650 fzoaddel 9664 fzosubel 9666 fzoend 9694 ssfzo12 9696 fzofzp1 9699 peano2fzor 9704 fzostep1 9709 iseqf1olemqk 9984 fzomaxdiflem 10606 fzo0dvdseq 11197 fzocongeq 11198 addmodlteqALT 11199 |
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