Theorem onuni 7743

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 7740	. 2 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
2		ssonuni 7735	. 2 ⊢ (𝐴 ∈ On → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
3	1, 2	mpd 15	1 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)

Syntax hints:   → wi 4   ∈ wcel 2114   ⊆ wss 3903  ∪ cuni 4865  Oncon0 6325

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 2709  ax-sep 5243  ax-pr 5379  ax-un 7690

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 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329

This theorem is referenced by:  onuninsuci  7792  oeeulem  8539  cnfcom3lem  9624  rankxpsuc  9806  dfac12lem2  10067  ttukeylem3  10433  r1limwun  10659  ontgval  36644  ordtoplem  36648  ordcmp  36660  1oequni2o  37617  rdgsucuni  37618  aomclem1  43405  omlimcl2  43593  onsucf1lem  43620  onsucf1olem  43621  onov0suclim  43625  dflim5  43680