Theorem onuni 7742

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 7739	. 2 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
2		ssonuni 7734	. 2 ⊢ (𝐴 ∈ On → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
3	1, 2	mpd 15	1 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)

Syntax hints:   → wi 4   ∈ wcel 2114   ⊆ wss 3889  ∪ cuni 4850  Oncon0 6323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327

This theorem is referenced by:  onuninsuci  7791  oeeulem  8537  cnfcom3lem  9624  rankxpsuc  9806  dfac12lem2  10067  ttukeylem3  10433  r1limwun  10659  ontgval  36613  ordtoplem  36617  ordcmp  36629  1oequni2o  37684  rdgsucuni  37685  aomclem1  43482  omlimcl2  43670  onsucf1lem  43697  onsucf1olem  43698  onov0suclim  43702  dflim5  43757