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Theorem onsucssi 7782
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 6433 . 2 Ord 𝐵
4 ordelsuc 7760 . 2 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))
51, 3, 4mp2an 691 1 (𝐴𝐵 ↔ suc 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2107  wss 3915  Ord word 6321  Oncon0 6322  suc csuc 6324
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-suc 6328
This theorem is referenced by:  omopthlem1  8610  rankval4  9810  rankc1  9813  rankc2  9814  rankxplim  9822  rankxplim3  9824  onsucsuccmpi  34944
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