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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnessex | Structured version Visualization version GIF version |
Description: If π΅ is finer than π΄ and π is an element of π΄, every point in π is an element of a subset of π which is in π΅. (Contributed by Jeff Hankins, 28-Sep-2009.) |
Ref | Expression |
---|---|
fnessex | β’ ((π΄Fneπ΅ β§ π β π΄ β§ π β π) β βπ₯ β π΅ (π β π₯ β§ π₯ β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnetg 35862 | . . 3 β’ (π΄Fneπ΅ β π΄ β (topGenβπ΅)) | |
2 | 1 | sselda 3982 | . 2 β’ ((π΄Fneπ΅ β§ π β π΄) β π β (topGenβπ΅)) |
3 | tg2 22888 | . 2 β’ ((π β (topGenβπ΅) β§ π β π) β βπ₯ β π΅ (π β π₯ β§ π₯ β π)) | |
4 | 2, 3 | stoic3 1770 | 1 β’ ((π΄Fneπ΅ β§ π β π΄ β§ π β π) β βπ₯ β π΅ (π β π₯ β§ π₯ β π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 β wcel 2098 βwrex 3067 β wss 3949 class class class wbr 5152 βcfv 6553 topGenctg 17426 Fnecfne 35853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-topgen 17432 df-fne 35854 |
This theorem is referenced by: fneint 35865 fnessref 35874 |
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