Theorem ssonunii 7802

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 7801	. 2 ⊢ (𝐴 ∈ V → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⊆ On → ∪ 𝐴 ∈ On)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  Vcvv 3453   ⊆ wss 3899  ∪ cuni 4869  Oncon0 6368

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-tr 5215  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-ord 6371  df-on 6372

This theorem is used by:  uniordint  7822  tz9.12lem2  9805  ttukeylem6  10602  nmulprop  37066  onsetreclem2  51013