Theorem onsucunifi

Description: The successor to the union of any non-empty, finite subset of ordinals is the union of the successors of the elements. (Contributed by RP, 12-Feb-2025)

	Ref	Expression
Assertion	onsucunifi	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → suc ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 suc 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		ordunifi	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → ∪ 𝐴 ∈ 𝐴 )
2		suceq	⊢ ( 𝑥 = ∪ 𝐴 → suc 𝑥 = suc ∪ 𝐴 )
3	2	ssiun2s	⊢ ( ∪ 𝐴 ∈ 𝐴 → suc ∪ 𝐴 ⊆ ∪ 𝑥 ∈ 𝐴 suc 𝑥 )
4	1 3	syl	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → suc ∪ 𝐴 ⊆ ∪ 𝑥 ∈ 𝐴 suc 𝑥 )
5		ssorduni	⊢ ( 𝐴 ⊆ On → Ord ∪ 𝐴 )
6		ordsuci	⊢ ( Ord ∪ 𝐴 → Ord suc ∪ 𝐴 )
7	5 6	syl	⊢ ( 𝐴 ⊆ On → Ord suc ∪ 𝐴 )
8		onsucuni	⊢ ( 𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴 )
9	8	sselda	⊢ ( ( 𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ suc ∪ 𝐴 )
10		ordsucss	⊢ ( Ord suc ∪ 𝐴 → ( 𝑥 ∈ suc ∪ 𝐴 → suc 𝑥 ⊆ suc ∪ 𝐴 ) )
11	10	imp	⊢ ( ( Ord suc ∪ 𝐴 ∧ 𝑥 ∈ suc ∪ 𝐴 ) → suc 𝑥 ⊆ suc ∪ 𝐴 )
12	7 9 11	syl2an2r	⊢ ( ( 𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴 ) → suc 𝑥 ⊆ suc ∪ 𝐴 )
13	12	iunssd	⊢ ( 𝐴 ⊆ On → ∪ 𝑥 ∈ 𝐴 suc 𝑥 ⊆ suc ∪ 𝐴 )
14	13	3ad2ant1	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → ∪ 𝑥 ∈ 𝐴 suc 𝑥 ⊆ suc ∪ 𝐴 )
15	4 14	eqssd	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → suc ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 suc 𝑥 )