Theorem ssonuni 7788

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 7787	. 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
2		uniexg 7751	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
3		elong 6375	. . 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2146  Vcvv 3458   ⊆ wss 3908  ∪ cuni 4877  Ord word 6366  Oncon0 6367

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-tr 5224  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-ord 6370  df-on 6371

This theorem is used by:  ssonunii  7789  onuni  7796  iunon  8335  onfununi  8337  oemapvali  9663  cardprclem  9984  carduni  9986  dfac12lem2  10147  ontgval  36983  onsupcl2  43993  onuniintrab  43994  onsupuni  43997  onsupcl3  44001  cantnfub2  44090