![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4250 | . . 3 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵..^𝐶) ≠ ∅) | |
2 | fzof 13030 | . . . . . 6 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | |
3 | 2 | fdmi 6498 | . . . . 5 ⊢ dom ..^ = (ℤ × ℤ) |
4 | 3 | ndmov 7312 | . . . 4 ⊢ (¬ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵..^𝐶) = ∅) |
5 | 4 | necon1ai 3014 | . . 3 ⊢ ((𝐵..^𝐶) ≠ ∅ → (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) |
7 | 6 | simpld 498 | 1 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 𝒫 cpw 4497 × cxp 5517 (class class class)co 7135 ℤcz 11969 ..^cfzo 13028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-neg 10862 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 |
This theorem is referenced by: elfzoelz 13033 elfzo2 13036 elfzole1 13041 elfzolt2 13042 elfzolt3 13043 elfzolt3b 13045 fzospliti 13064 fzoaddel 13085 elincfzoext 13090 fzosubel 13091 fzosubel3 13093 fzofzp1 13129 fzostep1 13148 fzomaxdiflem 14694 fzocongeq 15666 fzom1ne1 30550 caratheodorylem1 43165 |
Copyright terms: Public domain | W3C validator |