Theorem ssonunii 7801

Description: The union of a set of ordinal numbers is an ordinal number. Corollary 7N(d) of [Enderton] p. 193. (Contributed by NM, 20-Sep-2003.)

Hypothesis

Ref	Expression
ssonuni.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
ssonunii	⊢ (𝐴 ⊆ On → ∪ 𝐴 ∈ On)

Proof of Theorem ssonunii

Step	Hyp	Ref	Expression
1		ssonuni.1	. 2 ⊢ 𝐴 ∈ V
2		ssonuni 7800	. 2 ⊢ (𝐴 ∈ V → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⊆ On → ∪ 𝐴 ∈ On)

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3480   ⊆ wss 3951  ∪ cuni 4907  Oncon0 6384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388

This theorem is referenced by:  uniordint  7821  tz9.12lem2  9828  ttukeylem6  10554  onsetreclem2  49225