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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 4341 | . . 3 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵..^𝐶) ≠ ∅) | |
| 2 | fzof 13696 | . . . . . 6 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | |
| 3 | 2 | fdmi 6747 | . . . . 5 ⊢ dom ..^ = (ℤ × ℤ) |
| 4 | 3 | ndmov 7617 | . . . 4 ⊢ (¬ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵..^𝐶) = ∅) |
| 5 | 4 | necon1ai 2968 | . . 3 ⊢ ((𝐵..^𝐶) ≠ ∅ → (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) |
| 6 | 1, 5 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) |
| 7 | 6 | simpld 494 | 1 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 𝒫 cpw 4600 × cxp 5683 (class class class)co 7431 ℤcz 12613 ..^cfzo 13694 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-neg 11495 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 |
| This theorem is referenced by: elfzoelz 13699 elfzo2 13702 elfzole1 13707 elfzolt2 13708 elfzolt3 13709 elfzolt3b 13711 fzospliti 13731 fzoaddel 13756 elincfzoext 13762 fzosubel 13763 fzosubel3 13765 fzofzp1 13803 fzostep1 13822 fzomaxdiflem 15381 fzocongeq 16361 fzom1ne1 32803 caratheodorylem1 46541 gpgedgvtx1 48020 |
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