Theorem nadd1rabord

Description: The set of ordinals which have a natural sum less than some ordinal is an ordinal. (Contributed by RP, 20-Dec-2024)

	Ref	Expression
Assertion	nadd1rabord	⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → Ord { 𝑥 ∈ 𝐴 ∣ ( 𝑥 +no 𝐵 ) ∈ 𝐶 } )

Proof

Step	Hyp	Ref	Expression
1		ssrab2	⊢ { 𝑥 ∈ 𝐴 ∣ ( 𝑥 +no 𝐵 ) ∈ 𝐶 } ⊆ 𝐴
2		ordsson	⊢ ( Ord 𝐴 → 𝐴 ⊆ On )
3	2	3ad2ant1	⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → 𝐴 ⊆ On )
4	1 3	sstrid	⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → { 𝑥 ∈ 𝐴 ∣ ( 𝑥 +no 𝐵 ) ∈ 𝐶 } ⊆ On )
5		nadd1rabtr	⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → Tr { 𝑥 ∈ 𝐴 ∣ ( 𝑥 +no 𝐵 ) ∈ 𝐶 } )
6		dford5	⊢ ( Ord { 𝑥 ∈ 𝐴 ∣ ( 𝑥 +no 𝐵 ) ∈ 𝐶 } ↔ ( { 𝑥 ∈ 𝐴 ∣ ( 𝑥 +no 𝐵 ) ∈ 𝐶 } ⊆ On ∧ Tr { 𝑥 ∈ 𝐴 ∣ ( 𝑥 +no 𝐵 ) ∈ 𝐶 } ) )
7	4 5 6	sylanbrc	⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → Ord { 𝑥 ∈ 𝐴 ∣ ( 𝑥 +no 𝐵 ) ∈ 𝐶 } )