Theorem ssonunii 7769

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 7768	. 2 ⊢ (𝐴 ∈ V → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⊆ On → ∪ 𝐴 ∈ On)

Syntax hints:   → wi 4   ∈ wcel 2107  Vcvv 3457   ⊆ wss 3924  ∪ cuni 4880  Oncon0 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5263  ax-nul 5273  ax-pr 5399  ax-un 7723

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-pss 3944  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-br 5117  df-opab 5179  df-tr 5227  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6352  df-on 6353

This theorem is referenced by:  uniordint  7789  tz9.12lem2  9794  ttukeylem6  10520  onsetreclem2  49290