Step |
Hyp |
Ref |
Expression |
1 |
|
onfin2 |
⊢ ω = ( On ∩ Fin ) |
2 |
|
inss2 |
⊢ ( On ∩ Fin ) ⊆ Fin |
3 |
1 2
|
eqsstri |
⊢ ω ⊆ Fin |
4 |
|
2onn |
⊢ 2o ∈ ω |
5 |
3 4
|
sselii |
⊢ 2o ∈ Fin |
6 |
|
sdomdom |
⊢ ( 𝐴 ≺ 2o → 𝐴 ≼ 2o ) |
7 |
|
domfi |
⊢ ( ( 2o ∈ Fin ∧ 𝐴 ≼ 2o ) → 𝐴 ∈ Fin ) |
8 |
5 6 7
|
sylancr |
⊢ ( 𝐴 ≺ 2o → 𝐴 ∈ Fin ) |
9 |
|
id |
⊢ ( 𝐴 = ∅ → 𝐴 = ∅ ) |
10 |
|
0fin |
⊢ ∅ ∈ Fin |
11 |
9 10
|
eqeltrdi |
⊢ ( 𝐴 = ∅ → 𝐴 ∈ Fin ) |
12 |
|
1onn |
⊢ 1o ∈ ω |
13 |
3 12
|
sselii |
⊢ 1o ∈ Fin |
14 |
|
enfi |
⊢ ( 𝐴 ≈ 1o → ( 𝐴 ∈ Fin ↔ 1o ∈ Fin ) ) |
15 |
13 14
|
mpbiri |
⊢ ( 𝐴 ≈ 1o → 𝐴 ∈ Fin ) |
16 |
11 15
|
jaoi |
⊢ ( ( 𝐴 = ∅ ∨ 𝐴 ≈ 1o ) → 𝐴 ∈ Fin ) |
17 |
|
df2o3 |
⊢ 2o = { ∅ , 1o } |
18 |
17
|
eleq2i |
⊢ ( ( card ‘ 𝐴 ) ∈ 2o ↔ ( card ‘ 𝐴 ) ∈ { ∅ , 1o } ) |
19 |
|
fvex |
⊢ ( card ‘ 𝐴 ) ∈ V |
20 |
19
|
elpr |
⊢ ( ( card ‘ 𝐴 ) ∈ { ∅ , 1o } ↔ ( ( card ‘ 𝐴 ) = ∅ ∨ ( card ‘ 𝐴 ) = 1o ) ) |
21 |
18 20
|
bitri |
⊢ ( ( card ‘ 𝐴 ) ∈ 2o ↔ ( ( card ‘ 𝐴 ) = ∅ ∨ ( card ‘ 𝐴 ) = 1o ) ) |
22 |
21
|
a1i |
⊢ ( 𝐴 ∈ Fin → ( ( card ‘ 𝐴 ) ∈ 2o ↔ ( ( card ‘ 𝐴 ) = ∅ ∨ ( card ‘ 𝐴 ) = 1o ) ) ) |
23 |
|
cardnn |
⊢ ( 2o ∈ ω → ( card ‘ 2o ) = 2o ) |
24 |
4 23
|
ax-mp |
⊢ ( card ‘ 2o ) = 2o |
25 |
24
|
eleq2i |
⊢ ( ( card ‘ 𝐴 ) ∈ ( card ‘ 2o ) ↔ ( card ‘ 𝐴 ) ∈ 2o ) |
26 |
|
finnum |
⊢ ( 𝐴 ∈ Fin → 𝐴 ∈ dom card ) |
27 |
|
2on |
⊢ 2o ∈ On |
28 |
|
onenon |
⊢ ( 2o ∈ On → 2o ∈ dom card ) |
29 |
27 28
|
ax-mp |
⊢ 2o ∈ dom card |
30 |
|
cardsdom2 |
⊢ ( ( 𝐴 ∈ dom card ∧ 2o ∈ dom card ) → ( ( card ‘ 𝐴 ) ∈ ( card ‘ 2o ) ↔ 𝐴 ≺ 2o ) ) |
31 |
26 29 30
|
sylancl |
⊢ ( 𝐴 ∈ Fin → ( ( card ‘ 𝐴 ) ∈ ( card ‘ 2o ) ↔ 𝐴 ≺ 2o ) ) |
32 |
25 31
|
bitr3id |
⊢ ( 𝐴 ∈ Fin → ( ( card ‘ 𝐴 ) ∈ 2o ↔ 𝐴 ≺ 2o ) ) |
33 |
|
cardnueq0 |
⊢ ( 𝐴 ∈ dom card → ( ( card ‘ 𝐴 ) = ∅ ↔ 𝐴 = ∅ ) ) |
34 |
26 33
|
syl |
⊢ ( 𝐴 ∈ Fin → ( ( card ‘ 𝐴 ) = ∅ ↔ 𝐴 = ∅ ) ) |
35 |
|
cardnn |
⊢ ( 1o ∈ ω → ( card ‘ 1o ) = 1o ) |
36 |
12 35
|
ax-mp |
⊢ ( card ‘ 1o ) = 1o |
37 |
36
|
eqeq2i |
⊢ ( ( card ‘ 𝐴 ) = ( card ‘ 1o ) ↔ ( card ‘ 𝐴 ) = 1o ) |
38 |
|
finnum |
⊢ ( 1o ∈ Fin → 1o ∈ dom card ) |
39 |
13 38
|
ax-mp |
⊢ 1o ∈ dom card |
40 |
|
carden2 |
⊢ ( ( 𝐴 ∈ dom card ∧ 1o ∈ dom card ) → ( ( card ‘ 𝐴 ) = ( card ‘ 1o ) ↔ 𝐴 ≈ 1o ) ) |
41 |
26 39 40
|
sylancl |
⊢ ( 𝐴 ∈ Fin → ( ( card ‘ 𝐴 ) = ( card ‘ 1o ) ↔ 𝐴 ≈ 1o ) ) |
42 |
37 41
|
bitr3id |
⊢ ( 𝐴 ∈ Fin → ( ( card ‘ 𝐴 ) = 1o ↔ 𝐴 ≈ 1o ) ) |
43 |
34 42
|
orbi12d |
⊢ ( 𝐴 ∈ Fin → ( ( ( card ‘ 𝐴 ) = ∅ ∨ ( card ‘ 𝐴 ) = 1o ) ↔ ( 𝐴 = ∅ ∨ 𝐴 ≈ 1o ) ) ) |
44 |
22 32 43
|
3bitr3d |
⊢ ( 𝐴 ∈ Fin → ( 𝐴 ≺ 2o ↔ ( 𝐴 = ∅ ∨ 𝐴 ≈ 1o ) ) ) |
45 |
8 16 44
|
pm5.21nii |
⊢ ( 𝐴 ≺ 2o ↔ ( 𝐴 = ∅ ∨ 𝐴 ≈ 1o ) ) |