Theorem ssonuni 4472

Description: The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.)

Assertion

Ref	Expression
ssonuni	⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))

Proof of Theorem ssonuni

Step	Hyp	Ref	Expression
1		ssorduni 4471	. 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
2		uniexg 4424	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
3		elong 4358	. . 3 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ On ↔ Ord ∪ 𝐴))
4	2, 3	syl 14	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∈ On ↔ Ord ∪ 𝐴))
5	1, 4	syl5ibr 155	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))

Syntax hints:   → wi 4   ↔ wb 104   ∈ wcel 2141  Vcvv 2730   ⊆ wss 3121  ∪ cuni 3796  Ord word 4347  Oncon0 4348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353

This theorem is referenced by:  ssonunii  4473  onun2  4474  onuni  4478  iunon  6263