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Theorem onsucssi 7826
Description: A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 16-Sep-1995.)
Hypotheses
Ref Expression
onssi.1 𝐴 ∈ On
onsucssi.2 𝐵 ∈ On
Assertion
Ref Expression
onsucssi (𝐴𝐵 ↔ suc 𝐴𝐵)

Proof of Theorem onsucssi
StepHypRef Expression
1 onssi.1 . 2 𝐴 ∈ On
2 onsucssi.2 . . 3 𝐵 ∈ On
32onordi 6468 . 2 Ord 𝐵
4 ordelsuc 7804 . 2 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))
51, 3, 4mp2an 689 1 (𝐴𝐵 ↔ suc 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2098  wss 3943  Ord word 6356  Oncon0 6357  suc csuc 6359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361  df-suc 6363
This theorem is referenced by:  omopthlem1  8657  rankval4  9861  rankc1  9864  rankc2  9865  rankxplim  9873  rankxplim3  9875  cuteq1  27716  onsucsuccmpi  35835
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