Metamath Proof Explorer


Theorem ordpwsuc

Description: The collection of ordinals in the power class of an ordinal is its successor. (Contributed by NM, 30-Jan-2005)

Ref Expression
Assertion ordpwsuc ( Ord 𝐴 → ( 𝒫 𝐴 ∩ On ) = suc 𝐴 )

Proof

Step Hyp Ref Expression
1 elin ( 𝑥 ∈ ( 𝒫 𝐴 ∩ On ) ↔ ( 𝑥 ∈ 𝒫 𝐴𝑥 ∈ On ) )
2 velpw ( 𝑥 ∈ 𝒫 𝐴𝑥𝐴 )
3 2 anbi2ci ( ( 𝑥 ∈ 𝒫 𝐴𝑥 ∈ On ) ↔ ( 𝑥 ∈ On ∧ 𝑥𝐴 ) )
4 1 3 bitri ( 𝑥 ∈ ( 𝒫 𝐴 ∩ On ) ↔ ( 𝑥 ∈ On ∧ 𝑥𝐴 ) )
5 ordsssuc ( ( 𝑥 ∈ On ∧ Ord 𝐴 ) → ( 𝑥𝐴𝑥 ∈ suc 𝐴 ) )
6 5 expcom ( Ord 𝐴 → ( 𝑥 ∈ On → ( 𝑥𝐴𝑥 ∈ suc 𝐴 ) ) )
7 6 pm5.32d ( Ord 𝐴 → ( ( 𝑥 ∈ On ∧ 𝑥𝐴 ) ↔ ( 𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴 ) ) )
8 simpr ( ( 𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴 ) → 𝑥 ∈ suc 𝐴 )
9 ordsuc ( Ord 𝐴 ↔ Ord suc 𝐴 )
10 ordelon ( ( Ord suc 𝐴𝑥 ∈ suc 𝐴 ) → 𝑥 ∈ On )
11 10 ex ( Ord suc 𝐴 → ( 𝑥 ∈ suc 𝐴𝑥 ∈ On ) )
12 9 11 sylbi ( Ord 𝐴 → ( 𝑥 ∈ suc 𝐴𝑥 ∈ On ) )
13 12 ancrd ( Ord 𝐴 → ( 𝑥 ∈ suc 𝐴 → ( 𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴 ) ) )
14 8 13 impbid2 ( Ord 𝐴 → ( ( 𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴 ) ↔ 𝑥 ∈ suc 𝐴 ) )
15 7 14 bitrd ( Ord 𝐴 → ( ( 𝑥 ∈ On ∧ 𝑥𝐴 ) ↔ 𝑥 ∈ suc 𝐴 ) )
16 4 15 syl5bb ( Ord 𝐴 → ( 𝑥 ∈ ( 𝒫 𝐴 ∩ On ) ↔ 𝑥 ∈ suc 𝐴 ) )
17 16 eqrdv ( Ord 𝐴 → ( 𝒫 𝐴 ∩ On ) = suc 𝐴 )