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Theorem onunisuc 6496
Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) Generalize from onunisuci 6506. (Revised by BJ, 28-Dec-2024.)
Assertion
Ref Expression
onunisuc (𝐴 ∈ On → suc 𝐴 = 𝐴)

Proof of Theorem onunisuc
StepHypRef Expression
1 ontr 6495 . 2 (𝐴 ∈ On → Tr 𝐴)
2 unisucg 6464 . 2 (𝐴 ∈ On → (Tr 𝐴 suc 𝐴 = 𝐴))
31, 2mpbid 232 1 (𝐴 ∈ On → suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106   cuni 4912  Tr wtr 5265  Oncon0 6386  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-tr 5266  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392
This theorem is referenced by:  onunisuci  6506  nlimsuc  43431
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