Step | Hyp | Ref
| Expression |
1 | | isline.l |
. . . 4
β’ β€ =
(leβπΎ) |
2 | | isline.j |
. . . 4
β’ β¨ =
(joinβπΎ) |
3 | | isline.a |
. . . 4
β’ π΄ = (AtomsβπΎ) |
4 | | isline.n |
. . . 4
β’ π = (LinesβπΎ) |
5 | 1, 2, 3, 4 | lineset 38251 |
. . 3
β’ (πΎ β π· β π = {π₯ β£ βπ β π΄ βπ β π΄ (π β π β§ π₯ = {π β π΄ β£ π β€ (π β¨ π)})}) |
6 | 5 | eleq2d 2820 |
. 2
β’ (πΎ β π· β (π β π β π β {π₯ β£ βπ β π΄ βπ β π΄ (π β π β§ π₯ = {π β π΄ β£ π β€ (π β¨ π)})})) |
7 | 3 | fvexi 6860 |
. . . . . . . 8
β’ π΄ β V |
8 | 7 | rabex 5293 |
. . . . . . 7
β’ {π β π΄ β£ π β€ (π β¨ π)} β V |
9 | | eleq1 2822 |
. . . . . . 7
β’ (π = {π β π΄ β£ π β€ (π β¨ π)} β (π β V β {π β π΄ β£ π β€ (π β¨ π)} β V)) |
10 | 8, 9 | mpbiri 258 |
. . . . . 6
β’ (π = {π β π΄ β£ π β€ (π β¨ π)} β π β V) |
11 | 10 | adantl 483 |
. . . . 5
β’ ((π β π β§ π = {π β π΄ β£ π β€ (π β¨ π)}) β π β V) |
12 | 11 | a1i 11 |
. . . 4
β’ ((π β π΄ β§ π β π΄) β ((π β π β§ π = {π β π΄ β£ π β€ (π β¨ π)}) β π β V)) |
13 | 12 | rexlimivv 3193 |
. . 3
β’
(βπ β
π΄ βπ β π΄ (π β π β§ π = {π β π΄ β£ π β€ (π β¨ π)}) β π β V) |
14 | | eqeq1 2737 |
. . . . 5
β’ (π₯ = π β (π₯ = {π β π΄ β£ π β€ (π β¨ π)} β π = {π β π΄ β£ π β€ (π β¨ π)})) |
15 | 14 | anbi2d 630 |
. . . 4
β’ (π₯ = π β ((π β π β§ π₯ = {π β π΄ β£ π β€ (π β¨ π)}) β (π β π β§ π = {π β π΄ β£ π β€ (π β¨ π)}))) |
16 | 15 | 2rexbidv 3210 |
. . 3
β’ (π₯ = π β (βπ β π΄ βπ β π΄ (π β π β§ π₯ = {π β π΄ β£ π β€ (π β¨ π)}) β βπ β π΄ βπ β π΄ (π β π β§ π = {π β π΄ β£ π β€ (π β¨ π)}))) |
17 | 13, 16 | elab3 3642 |
. 2
β’ (π β {π₯ β£ βπ β π΄ βπ β π΄ (π β π β§ π₯ = {π β π΄ β£ π β€ (π β¨ π)})} β βπ β π΄ βπ β π΄ (π β π β§ π = {π β π΄ β£ π β€ (π β¨ π)})) |
18 | 6, 17 | bitrdi 287 |
1
β’ (πΎ β π· β (π β π β βπ β π΄ βπ β π΄ (π β π β§ π = {π β π΄ β£ π β€ (π β¨ π)}))) |