| Step | Hyp | Ref
| Expression |
|---|
| 1 | | 2onn 8637 |
. . . . . . . . . 10
β’
2o β Ο |
| 2 | | nnfi 9163 |
. . . . . . . . . 10
β’
(2o β Ο β 2o β
Fin) |
| 3 | 1, 2 | ax-mp 5 |
. . . . . . . . 9
β’
2o β Fin |
| 4 | | enfi 9186 |
. . . . . . . . 9
β’ (π β 2o β
(π β Fin β
2o β Fin)) |
| 5 | 3, 4 | mpbiri 258 |
. . . . . . . 8
β’ (π β 2o β
π β
Fin) |
| 6 | 5 | adantl 481 |
. . . . . . 7
β’ ((π β π β§ π β 2o) β π β Fin) |
| 7 | | diffi 9175 |
. . . . . . 7
β’ (π β Fin β (π β {π}) β Fin) |
| 8 | 6, 7 | syl 17 |
. . . . . 6
β’ ((π β π β§ π β 2o) β (π β {π}) β Fin) |
| 9 | 8 | cardidd 10540 |
. . . . 5
β’ ((π β π β§ π β 2o) β
(cardβ(π β
{π})) β (π β {π})) |
| 10 | 9 | ensymd 8997 |
. . . 4
β’ ((π β π β§ π β 2o) β (π β {π}) β (cardβ(π β {π}))) |
| 11 | | simpl 482 |
. . . . . . 7
β’ ((π β π β§ π β 2o) β π β π) |
| 12 | | dif1card 10001 |
. . . . . . 7
β’ ((π β Fin β§ π β π) β (cardβπ) = suc (cardβ(π β {π}))) |
| 13 | 6, 11, 12 | syl2anc 583 |
. . . . . 6
β’ ((π β π β§ π β 2o) β
(cardβπ) = suc
(cardβ(π β
{π}))) |
| 14 | | cardennn 9974 |
. . . . . . . . 9
β’ ((π β 2o β§
2o β Ο) β (cardβπ) = 2o) |
| 15 | 1, 14 | mpan2 688 |
. . . . . . . 8
β’ (π β 2o β
(cardβπ) =
2o) |
| 16 | | df-2o 8462 |
. . . . . . . 8
β’
2o = suc 1o |
| 17 | 15, 16 | eqtrdi 2780 |
. . . . . . 7
β’ (π β 2o β
(cardβπ) = suc
1o) |
| 18 | 17 | adantl 481 |
. . . . . 6
β’ ((π β π β§ π β 2o) β
(cardβπ) = suc
1o) |
| 19 | 13, 18 | eqtr3d 2766 |
. . . . 5
β’ ((π β π β§ π β 2o) β suc
(cardβ(π β
{π})) = suc
1o) |
| 20 | | suc11reg 9610 |
. . . . 5
β’ (suc
(cardβ(π β
{π})) = suc 1o
β (cardβ(π
β {π})) =
1o) |
| 21 | 19, 20 | sylib 217 |
. . . 4
β’ ((π β π β§ π β 2o) β
(cardβ(π β
{π})) =
1o) |
| 22 | 10, 21 | breqtrd 5164 |
. . 3
β’ ((π β π β§ π β 2o) β (π β {π}) β 1o) |
| 23 | | en1 9017 |
. . 3
β’ ((π β {π}) β 1o β βπ₯(π β {π}) = {π₯}) |
| 24 | 22, 23 | sylib 217 |
. 2
β’ ((π β π β§ π β 2o) β βπ₯(π β {π}) = {π₯}) |
| 25 | | simpr 484 |
. . . . 5
β’ (((π β π β§ π β 2o) β§ (π β {π}) = {π₯}) β (π β {π}) = {π₯}) |
| 26 | 25 | unieqd 4912 |
. . . 4
β’ (((π β π β§ π β 2o) β§ (π β {π}) = {π₯}) β βͺ (π β {π}) = βͺ {π₯}) |
| 27 | | unisnv 4921 |
. . . 4
β’ βͺ {π₯}
= π₯ |
| 28 | 26, 27 | eqtrdi 2780 |
. . 3
β’ (((π β π β§ π β 2o) β§ (π β {π}) = {π₯}) β βͺ (π β {π}) = π₯) |
| 29 | | difssd 4124 |
. . . . 5
β’ (((π β π β§ π β 2o) β§ (π β {π}) = {π₯}) β (π β {π}) β π) |
| 30 | 25, 29 | eqsstrrd 4013 |
. . . 4
β’ (((π β π β§ π β 2o) β§ (π β {π}) = {π₯}) β {π₯} β π) |
| 31 | | vsnid 4657 |
. . . 4
β’ π₯ β {π₯} |
| 32 | | ssel2 3969 |
. . . 4
β’ (({π₯} β π β§ π₯ β {π₯}) β π₯ β π) |
| 33 | 30, 31, 32 | sylancl 585 |
. . 3
β’ (((π β π β§ π β 2o) β§ (π β {π}) = {π₯}) β π₯ β π) |
| 34 | 28, 33 | eqeltrd 2825 |
. 2
β’ (((π β π β§ π β 2o) β§ (π β {π}) = {π₯}) β βͺ (π β {π}) β π) |
| 35 | 24, 34 | exlimddv 1930 |
1
β’ ((π β π β§ π β 2o) β βͺ (π
β {π}) β π) |
