Theorem ssonuni 7780

Description: The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. Lemma 2.7 of [Schloeder] p. 4. (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 7779	. 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
2		uniexg 7740	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
3		elong 6370	. . 3 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ On ↔ Ord ∪ 𝐴))
4	2, 3	syl 18	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∈ On ↔ Ord ∪ 𝐴))
5	1, 4	imbitrrid 249	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3906  ∪ cuni 4873  Ord word 6361  Oncon0 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-tr 5220  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6365  df-on 6366

This theorem is referenced by:  ssonunii  7781  onuni  7788  iunon  8327  onfununi  8329  oemapvali  9654  cardprclem  9966  carduni  9968  dfac12lem2  10129  ontgval  36920  onsupcl2  43932  onuniintrab  43933  onsupuni  43936  onsupcl3  43940  cantnfub2  44029