Theorem onuni 7721

Description: The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)

Assertion

Ref	Expression
onuni	⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)

Proof of Theorem onuni

Step	Hyp	Ref	Expression
1		onss 7718	. 2 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
2		ssonuni 7713	. 2 ⊢ (𝐴 ∈ On → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
3	1, 2	mpd 15	1 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)

Syntax hints:   → wi 4   ∈ wcel 2111   ⊆ wss 3902  ∪ cuni 4859  Oncon0 6306

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

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 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-tr 5199  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-ord 6309  df-on 6310

This theorem is referenced by:  onuninsuci  7770  oeeulem  8516  cnfcom3lem  9593  rankxpsuc  9772  dfac12lem2  10033  ttukeylem3  10399  r1limwun  10624  ontgval  36464  ordtoplem  36468  ordcmp  36480  1oequni2o  37401  rdgsucuni  37402  aomclem1  43086  omlimcl2  43274  onsucf1lem  43301  onsucf1olem  43302  onov0suclim  43306  dflim5  43361