Theorem ssonunii 7764

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 7763	. 2 ⊢ (𝐴 ∈ V → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⊆ On → ∪ 𝐴 ∈ On)

Syntax hints:   → wi 4   ∈ wcel 2098  Vcvv 3468   ⊆ wss 3943  ∪ cuni 4902  Oncon0 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361

This theorem is referenced by:  uniordint  7785  tz9.12lem2  9782  ttukeylem6  10508  onsetreclem2  48007