Metamath Proof Explorer


Theorem bday0s

Description: Calculate the birthday of surreal zero. (Contributed by Scott Fenton, 7-Aug-2024)

Ref Expression
Assertion bday0s ( bday ‘ 0s ) = ∅

Proof

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 ) = ∅