Theorem ssonuni 7735

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 7734	. 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
2		uniexg 7695	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
3		elong 6333	. . 3 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ On ↔ Ord ∪ 𝐴))
4	2, 3	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∈ On ↔ Ord ∪ 𝐴))
5	1, 4	imbitrrid 246	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2114  Vcvv 3442   ⊆ wss 3903  ∪ cuni 4865  Ord word 6324  Oncon0 6325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329

This theorem is referenced by:  ssonunii  7736  onuni  7743  iunon  8281  onfununi  8283  oemapvali  9605  cardprclem  9903  carduni  9905  dfac12lem2  10067  ontgval  36644  onsupcl2  43576  onuniintrab  43577  onsupuni  43580  onsupcl3  43584  cantnfub2  43673