Theorem ssonuni 7725

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 7724	. 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
2		uniexg 7685	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
3		elong 6325	. . 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 2113  Vcvv 3440   ⊆ wss 3901  ∪ cuni 4863  Ord word 6316  Oncon0 6317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321

This theorem is referenced by:  ssonunii  7726  onuni  7733  iunon  8271  onfununi  8273  oemapvali  9593  cardprclem  9891  carduni  9893  dfac12lem2  10055  ontgval  36625  onsupcl2  43463  onuniintrab  43464  onsupuni  43467  onsupcl3  43471  cantnfub2  43560