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Mirrors > Home > MPE Home > Th. List > ist1 | Structured version Visualization version GIF version |
Description: The predicate "is a T1 space". (Contributed by FL, 18-Jun-2007.) |
Ref | Expression |
---|---|
ist0.1 | β’ π = βͺ π½ |
Ref | Expression |
---|---|
ist1 | β’ (π½ β Fre β (π½ β Top β§ βπ β π {π} β (Clsdβπ½))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 4863 | . . . 4 β’ (π₯ = π½ β βͺ π₯ = βͺ π½) | |
2 | ist0.1 | . . . 4 β’ π = βͺ π½ | |
3 | 1, 2 | eqtr4di 2794 | . . 3 β’ (π₯ = π½ β βͺ π₯ = π) |
4 | fveq2 6825 | . . . 4 β’ (π₯ = π½ β (Clsdβπ₯) = (Clsdβπ½)) | |
5 | 4 | eleq2d 2822 | . . 3 β’ (π₯ = π½ β ({π} β (Clsdβπ₯) β {π} β (Clsdβπ½))) |
6 | 3, 5 | raleqbidv 3315 | . 2 β’ (π₯ = π½ β (βπ β βͺ π₯{π} β (Clsdβπ₯) β βπ β π {π} β (Clsdβπ½))) |
7 | df-t1 22571 | . 2 β’ Fre = {π₯ β Top β£ βπ β βͺ π₯{π} β (Clsdβπ₯)} | |
8 | 6, 7 | elrab2 3637 | 1 β’ (π½ β Fre β (π½ β Top β§ βπ β π {π} β (Clsdβπ½))) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 396 = wceq 1540 β wcel 2105 βwral 3061 {csn 4573 βͺ cuni 4852 βcfv 6479 Topctop 22148 Clsdccld 22273 Frect1 22564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3062 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-iota 6431 df-fv 6487 df-t1 22571 |
This theorem is referenced by: t1sncld 22583 t1ficld 22584 t1top 22587 ist1-2 22604 cnt1 22607 ordtt1 22636 qtopt1 32083 zarmxt1 32128 onint1 34734 |
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