Theorem limexissupab

Description: An ordinal which is a limit ordinal is equal to the supremum of the class of all its elements. Lemma 2.13 of Schloeder p. 5. (Contributed by RP, 27-Jan-2025)

	Ref	Expression
Assertion	limexissupab	⊢ ( ( Lim 𝐴 ∧ 𝐴 ∈ 𝑉 ) → 𝐴 = sup ( { 𝑥 ∣ 𝑥 ∈ 𝐴 } , On , E ) )

Proof

Step	Hyp	Ref	Expression
1		limuni	⊢ ( Lim 𝐴 → 𝐴 = ∪ 𝐴 )
2	1	adantr	⊢ ( ( Lim 𝐴 ∧ 𝐴 ∈ 𝑉 ) → 𝐴 = ∪ 𝐴 )
3		limord	⊢ ( Lim 𝐴 → Ord 𝐴 )
4		ordsson	⊢ ( Ord 𝐴 → 𝐴 ⊆ On )
5	3 4	syl	⊢ ( Lim 𝐴 → 𝐴 ⊆ On )
6		onsupuni	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) → sup ( 𝐴 , On , E ) = ∪ 𝐴 )
7	5 6	sylan	⊢ ( ( Lim 𝐴 ∧ 𝐴 ∈ 𝑉 ) → sup ( 𝐴 , On , E ) = ∪ 𝐴 )
8		abid1	⊢ 𝐴 = { 𝑥 ∣ 𝑥 ∈ 𝐴 }
9		supeq1	⊢ ( 𝐴 = { 𝑥 ∣ 𝑥 ∈ 𝐴 } → sup ( 𝐴 , On , E ) = sup ( { 𝑥 ∣ 𝑥 ∈ 𝐴 } , On , E ) )
10	8 9	mp1i	⊢ ( ( Lim 𝐴 ∧ 𝐴 ∈ 𝑉 ) → sup ( 𝐴 , On , E ) = sup ( { 𝑥 ∣ 𝑥 ∈ 𝐴 } , On , E ) )
11	2 7 10	3eqtr2d	⊢ ( ( Lim 𝐴 ∧ 𝐴 ∈ 𝑉 ) → 𝐴 = sup ( { 𝑥 ∣ 𝑥 ∈ 𝐴 } , On , E ) )