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Mirrors > Home > MPE Home > Th. List > Mathboxes > gneispb | Structured version Visualization version GIF version |
Description: Given a neighborhood π of π, each subset of the neighborhood space containing this neighborhood is also a neighborhood of π. Axiom B of Seifert and Threlfall. (Contributed by RP, 5-Apr-2021.) |
Ref | Expression |
---|---|
gneispace.x | β’ π = βͺ π½ |
Ref | Expression |
---|---|
gneispb | β’ ((π½ β Top β§ π β π β§ π β ((neiβπ½)β{π})) β βπ β π« π(π β π β π β ((neiβπ½)β{π}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpb 1150 | . . . . 5 β’ ((π½ β Top β§ π β π β§ π β ((neiβπ½)β{π})) β (π½ β Top β§ π β ((neiβπ½)β{π}))) | |
2 | 1 | ad2antrr 725 | . . . 4 β’ ((((π½ β Top β§ π β π β§ π β ((neiβπ½)β{π})) β§ π β π« π) β§ π β π ) β (π½ β Top β§ π β ((neiβπ½)β{π}))) |
3 | simpr 486 | . . . 4 β’ ((((π½ β Top β§ π β π β§ π β ((neiβπ½)β{π})) β§ π β π« π) β§ π β π ) β π β π ) | |
4 | simplr 768 | . . . . 5 β’ ((((π½ β Top β§ π β π β§ π β ((neiβπ½)β{π})) β§ π β π« π) β§ π β π ) β π β π« π) | |
5 | 4 | elpwid 4573 | . . . 4 β’ ((((π½ β Top β§ π β π β§ π β ((neiβπ½)β{π})) β§ π β π« π) β§ π β π ) β π β π) |
6 | gneispace.x | . . . . 5 β’ π = βͺ π½ | |
7 | 6 | ssnei2 22490 | . . . 4 β’ (((π½ β Top β§ π β ((neiβπ½)β{π})) β§ (π β π β§ π β π)) β π β ((neiβπ½)β{π})) |
8 | 2, 3, 5, 7 | syl12anc 836 | . . 3 β’ ((((π½ β Top β§ π β π β§ π β ((neiβπ½)β{π})) β§ π β π« π) β§ π β π ) β π β ((neiβπ½)β{π})) |
9 | 8 | exp31 421 | . 2 β’ ((π½ β Top β§ π β π β§ π β ((neiβπ½)β{π})) β (π β π« π β (π β π β π β ((neiβπ½)β{π})))) |
10 | 9 | ralrimiv 3139 | 1 β’ ((π½ β Top β§ π β π β§ π β ((neiβπ½)β{π})) β βπ β π« π(π β π β π β ((neiβπ½)β{π}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 β wss 3914 π« cpw 4564 {csn 4590 βͺ cuni 4869 βcfv 6500 Topctop 22265 neicnei 22471 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-top 22266 df-nei 22472 |
This theorem is referenced by: (None) |
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