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Mirrors > Home > MPE Home > Th. List > elfzoel1 | Structured version Visualization version 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 | ne0i 3880 | . . 3 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵..^𝐶) ≠ ∅) | |
2 | fzof 12336 | . . . . . 6 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | |
3 | 2 | fdmi 5965 | . . . . 5 ⊢ dom ..^ = (ℤ × ℤ) |
4 | 3 | ndmov 6716 | . . . 4 ⊢ (¬ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵..^𝐶) = ∅) |
5 | 4 | necon1ai 2809 | . . 3 ⊢ ((𝐵..^𝐶) ≠ ∅ → (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) |
7 | 6 | simpld 474 | 1 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 𝒫 cpw 4108 × cxp 5036 (class class class)co 6549 ℤcz 11254 ..^cfzo 12334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-neg 10148 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 |
This theorem is referenced by: elfzoelz 12339 elfzo2 12342 elfzole1 12347 elfzolt2 12348 elfzolt3 12349 elfzolt3b 12351 fzospliti 12369 fzoaddel 12388 elincfzoext 12393 fzosubel 12394 fzosubel3 12396 fzofzp1 12431 fzostep1 12446 fzomaxdiflem 13930 fzocongeq 14884 caratheodorylem1 39416 |
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