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Theorem onsucssi 7026
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 5820 . 2 Ord 𝐵
4 ordelsuc 7005 . 2 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))
51, 3, 4mp2an 707 1 (𝐴𝐵 ↔ suc 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 1988  wss 3567  Ord word 5710  Oncon0 5711  suc csuc 5713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-ord 5714  df-on 5715  df-suc 5717
This theorem is referenced by:  omopthlem1  7720  rankval4  8715  rankc1  8718  rankc2  8719  rankxplim  8727  rankxplim3  8729  onsucsuccmpi  32417
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