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Theorem onsucssi 7273
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 6043 . 2 Ord 𝐵
4 ordelsuc 7252 . 2 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))
51, 3, 4mp2an 684 1 (𝐴𝐵 ↔ suc 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wcel 2157  wss 3767  Ord word 5938  Oncon0 5939  suc csuc 5941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-tr 4944  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-ord 5942  df-on 5943  df-suc 5945
This theorem is referenced by:  omopthlem1  7973  rankval4  8978  rankc1  8981  rankc2  8982  rankxplim  8990  rankxplim3  8992  onsucsuccmpi  32942
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