Step |
Hyp |
Ref |
Expression |
1 |
|
df-0s |
⊢ 0s = ( ∅ |s ∅ ) |
2 |
1
|
fveq2i |
⊢ ( bday ‘ 0s ) = ( bday ‘ ( ∅ |s ∅ ) ) |
3 |
|
0elpw |
⊢ ∅ ∈ 𝒫 No |
4 |
|
nulssgt |
⊢ ( ∅ ∈ 𝒫 No → ∅ <<s ∅ ) |
5 |
|
scutbday |
⊢ ( ∅ <<s ∅ → ( bday ‘ ( ∅ |s ∅ ) ) = ∩ ( bday “ { 𝑥 ∈ No ∣ ( ∅ <<s { 𝑥 } ∧ { 𝑥 } <<s ∅ ) } ) ) |
6 |
3 4 5
|
mp2b |
⊢ ( bday ‘ ( ∅ |s ∅ ) ) = ∩ ( bday “ { 𝑥 ∈ No ∣ ( ∅ <<s { 𝑥 } ∧ { 𝑥 } <<s ∅ ) } ) |
7 |
2 6
|
eqtri |
⊢ ( bday ‘ 0s ) = ∩ ( bday “ { 𝑥 ∈ No ∣ ( ∅ <<s { 𝑥 } ∧ { 𝑥 } <<s ∅ ) } ) |
8 |
|
snelpwi |
⊢ ( 𝑥 ∈ No → { 𝑥 } ∈ 𝒫 No ) |
9 |
|
nulsslt |
⊢ ( { 𝑥 } ∈ 𝒫 No → ∅ <<s { 𝑥 } ) |
10 |
|
nulssgt |
⊢ ( { 𝑥 } ∈ 𝒫 No → { 𝑥 } <<s ∅ ) |
11 |
9 10
|
jca |
⊢ ( { 𝑥 } ∈ 𝒫 No → ( ∅ <<s { 𝑥 } ∧ { 𝑥 } <<s ∅ ) ) |
12 |
8 11
|
syl |
⊢ ( 𝑥 ∈ No → ( ∅ <<s { 𝑥 } ∧ { 𝑥 } <<s ∅ ) ) |
13 |
12
|
rabeqc |
⊢ { 𝑥 ∈ No ∣ ( ∅ <<s { 𝑥 } ∧ { 𝑥 } <<s ∅ ) } = No |
14 |
|
bdaydm |
⊢ dom bday = No |
15 |
13 14
|
eqtr4i |
⊢ { 𝑥 ∈ No ∣ ( ∅ <<s { 𝑥 } ∧ { 𝑥 } <<s ∅ ) } = dom bday |
16 |
15
|
imaeq2i |
⊢ ( bday “ { 𝑥 ∈ No ∣ ( ∅ <<s { 𝑥 } ∧ { 𝑥 } <<s ∅ ) } ) = ( bday “ dom bday ) |
17 |
|
imadmrn |
⊢ ( bday “ dom bday ) = ran bday |
18 |
|
bdayrn |
⊢ ran bday = On |
19 |
16 17 18
|
3eqtri |
⊢ ( bday “ { 𝑥 ∈ No ∣ ( ∅ <<s { 𝑥 } ∧ { 𝑥 } <<s ∅ ) } ) = On |
20 |
19
|
inteqi |
⊢ ∩ ( bday “ { 𝑥 ∈ No ∣ ( ∅ <<s { 𝑥 } ∧ { 𝑥 } <<s ∅ ) } ) = ∩ On |
21 |
|
inton |
⊢ ∩ On = ∅ |
22 |
7 20 21
|
3eqtri |
⊢ ( bday ‘ 0s ) = ∅ |