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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 4268 | . . 3 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵..^𝐶) ≠ ∅) | |
2 | fzof 13384 | . . . . . 6 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | |
3 | 2 | fdmi 6612 | . . . . 5 ⊢ dom ..^ = (ℤ × ℤ) |
4 | 3 | ndmov 7456 | . . . 4 ⊢ (¬ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵..^𝐶) = ∅) |
5 | 4 | necon1ai 2971 | . . 3 ⊢ ((𝐵..^𝐶) ≠ ∅ → (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) |
7 | 6 | simpld 495 | 1 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ≠ wne 2943 ∅c0 4256 𝒫 cpw 4533 × cxp 5587 (class class class)co 7275 ℤcz 12319 ..^cfzo 13382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-neg 11208 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 |
This theorem is referenced by: elfzoelz 13387 elfzo2 13390 elfzole1 13395 elfzolt2 13396 elfzolt3 13397 elfzolt3b 13399 fzospliti 13419 fzoaddel 13440 elincfzoext 13445 fzosubel 13446 fzosubel3 13448 fzofzp1 13484 fzostep1 13503 fzomaxdiflem 15054 fzocongeq 16033 fzom1ne1 31122 caratheodorylem1 44064 |
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