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Mirrors > Home > MPE Home > Th. List > mreuni | Structured version Visualization version GIF version |
Description: Since the entire base set of a Moore collection is the greatest element of it, the base set can be recovered from a Moore collection by set union. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Ref | Expression |
---|---|
mreuni | β’ (πΆ β (Mooreβπ) β βͺ πΆ = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mre1cl 17542 | . 2 β’ (πΆ β (Mooreβπ) β π β πΆ) | |
2 | mresspw 17540 | . 2 β’ (πΆ β (Mooreβπ) β πΆ β π« π) | |
3 | elpwuni 5107 | . . 3 β’ (π β πΆ β (πΆ β π« π β βͺ πΆ = π)) | |
4 | 3 | biimpa 475 | . 2 β’ ((π β πΆ β§ πΆ β π« π) β βͺ πΆ = π) |
5 | 1, 2, 4 | syl2anc 582 | 1 β’ (πΆ β (Mooreβπ) β βͺ πΆ = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 β wss 3947 π« cpw 4601 βͺ cuni 4907 βcfv 6542 Moorecmre 17530 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-mre 17534 |
This theorem is referenced by: mreunirn 17549 mrcfval 17556 mrcssv 17562 mrisval 17578 mrelatlub 18519 mreclatBAD 18520 mreuniss 47619 clduni 47620 mreclat 47709 |
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