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Theorem onuniorsuci 7364
 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.)
Hypothesis
Ref Expression
onssi.1 𝐴 ∈ On
Assertion
Ref Expression
onuniorsuci (𝐴 = 𝐴𝐴 = suc 𝐴)

Proof of Theorem onuniorsuci
StepHypRef Expression
1 onssi.1 . . 3 𝐴 ∈ On
21onordi 6127 . 2 Ord 𝐴
3 orduniorsuc 7355 . 2 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
42, 3ax-mp 5 1 (𝐴 = 𝐴𝐴 = suc 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 833   = wceq 1507   ∈ wcel 2048  ∪ cuni 4706  Ord word 6022  Oncon0 6023  suc csuc 6025 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pr 5180  ax-un 7273 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-tr 5025  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-ord 6026  df-on 6027  df-suc 6029 This theorem is referenced by:  onuninsuci  7365
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