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Theorem onuniorsuc 7777
Description: An ordinal number is either its own union (if zero or a limit ordinal) or the successor of its union. (Contributed by NM, 13-Jun-1994.) Put in closed form. (Revised by BJ, 11-Jan-2025.)
Assertion
Ref Expression
onuniorsuc (𝐴 ∈ On → (𝐴 = 𝐴𝐴 = suc 𝐴))

Proof of Theorem onuniorsuc
StepHypRef Expression
1 eloni 6332 . 2 (𝐴 ∈ On → Ord 𝐴)
2 orduniorsuc 7770 . 2 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
31, 2syl 17 1 (𝐴 ∈ On → (𝐴 = 𝐴𝐴 = suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 846   = wceq 1542  wcel 2107   cuni 4870  Ord word 6321  Oncon0 6322  suc csuc 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-suc 6328
This theorem is referenced by:  onuniorsuciOLD  7780  onuninsuci  7781  onsucf1olem  41634
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