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Theorem onunisuci 6179
 Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)
Hypothesis
Ref Expression
on.1 𝐴 ∈ On
Assertion
Ref Expression
onunisuci suc 𝐴 = 𝐴

Proof of Theorem onunisuci
StepHypRef Expression
1 on.1 . . 3 𝐴 ∈ On
21ontrci 6171 . 2 Tr 𝐴
31elexi 3456 . . 3 𝐴 ∈ V
43unisuc 6142 . 2 (Tr 𝐴 suc 𝐴 = 𝐴)
52, 4mpbi 231 1 suc 𝐴 = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1522   ∈ wcel 2081  ∪ cuni 4745  Tr wtr 5063  Oncon0 6066  suc csuc 6068 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-v 3439  df-un 3864  df-in 3866  df-ss 3874  df-sn 4473  df-pr 4475  df-uni 4746  df-tr 5064  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-ord 6069  df-on 6070  df-suc 6072 This theorem is referenced by:  rankuni  9138  onsucconni  33394  onsucsuccmpi  33400  finxp1o  34204
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