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Theorem onunisuci 6284
 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 6276 . 2 Tr 𝐴
31elexi 3430 . . 3 𝐴 ∈ V
43unisuc 6246 . 2 (Tr 𝐴 suc 𝐴 = 𝐴)
52, 4mpbi 233 1 suc 𝐴 = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1539   ∈ wcel 2112  ∪ cuni 4799  Tr wtr 5139  Oncon0 6170  suc csuc 6172 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-v 3412  df-un 3864  df-in 3866  df-ss 3876  df-sn 4524  df-pr 4526  df-uni 4800  df-tr 5140  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-ord 6173  df-on 6174  df-suc 6176 This theorem is referenced by:  rankuni  9318  onsucconni  34168  onsucsuccmpi  34174  finxp1o  35082
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