Theorem onuni 7788

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 7785	. 2 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
2		ssonuni 7780	. 2 ⊢ (𝐴 ∈ On → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
3	1, 2	mpd 16	1 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)

Syntax hints:   → wi 4   ∈ wcel 2143   ⊆ wss 3906  ∪ cuni 4873  Oncon0 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-tr 5220  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6365  df-on 6366

This theorem is referenced by:  onuninsuci  7837  oeeulem  8588  cnfcom3lem  9673  rankxpsuc  9855  dfac12lem2  10129  ttukeylem3  10496  r1limwun  10722  ontgval  36920  ordtoplem  36924  ordcmp  36936  1oequni2o  37992  rdgsucuni  37993  aomclem1  43761  omlimcl2  43949  onsucf1lem  43976  onsucf1olem  43977  onov0suclim  43981  dflim5  44036