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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 4295 | . . 3 β’ (π΄ β (π΅..^πΆ) β (π΅..^πΆ) β β ) | |
2 | fzof 13570 | . . . . . 6 β’ ..^:(β€ Γ β€)βΆπ« β€ | |
3 | 2 | fdmi 6681 | . . . . 5 β’ dom ..^ = (β€ Γ β€) |
4 | 3 | ndmov 7539 | . . . 4 β’ (Β¬ (π΅ β β€ β§ πΆ β β€) β (π΅..^πΆ) = β ) |
5 | 4 | necon1ai 2972 | . . 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 496 | 1 β’ (π΄ β (π΅..^πΆ) β π΅ β β€) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β wcel 2107 β wne 2944 β c0 4283 π« cpw 4561 Γ cxp 5632 (class class class)co 7358 β€cz 12500 ..^cfzo 13568 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-neg 11389 df-z 12501 df-uz 12765 df-fz 13426 df-fzo 13569 |
This theorem is referenced by: elfzoelz 13573 elfzo2 13576 elfzole1 13581 elfzolt2 13582 elfzolt3 13583 elfzolt3b 13585 fzospliti 13605 fzoaddel 13626 elincfzoext 13631 fzosubel 13632 fzosubel3 13634 fzofzp1 13670 fzostep1 13689 fzomaxdiflem 15228 fzocongeq 16207 fzom1ne1 31707 caratheodorylem1 44774 |
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