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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 | β’ (π΄ β (π΅..^πΆ) β π΅ β β€) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 4334 | . . 3 β’ (π΄ β (π΅..^πΆ) β (π΅..^πΆ) β β ) | |
2 | fzof 13631 | . . . . . 6 β’ ..^:(β€ Γ β€)βΆπ« β€ | |
3 | 2 | fdmi 6729 | . . . . 5 β’ dom ..^ = (β€ Γ β€) |
4 | 3 | ndmov 7593 | . . . 4 β’ (Β¬ (π΅ β β€ β§ πΆ β β€) β (π΅..^πΆ) = β ) |
5 | 4 | necon1ai 2968 | . . 3 β’ ((π΅..^πΆ) β β β (π΅ β β€ β§ πΆ β β€)) |
6 | 1, 5 | syl 17 | . 2 β’ (π΄ β (π΅..^πΆ) β (π΅ β β€ β§ πΆ β β€)) |
7 | 6 | simpld 495 | 1 β’ (π΄ β (π΅..^πΆ) β π΅ β β€) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β wcel 2106 β wne 2940 β c0 4322 π« cpw 4602 Γ cxp 5674 (class class class)co 7411 β€cz 12560 ..^cfzo 13629 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-neg 11449 df-z 12561 df-uz 12825 df-fz 13487 df-fzo 13630 |
This theorem is referenced by: elfzoelz 13634 elfzo2 13637 elfzole1 13642 elfzolt2 13643 elfzolt3 13644 elfzolt3b 13646 fzospliti 13666 fzoaddel 13687 elincfzoext 13692 fzosubel 13693 fzosubel3 13695 fzofzp1 13731 fzostep1 13750 fzomaxdiflem 15291 fzocongeq 16269 fzom1ne1 32050 caratheodorylem1 45321 |
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