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Theorem onsucssi 4252
 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
onsucssi.1 𝐴 ∈ On
onsucssi.2 𝐵 ∈ On
Assertion
Ref Expression
onsucssi (𝐴𝐵 ↔ suc 𝐴𝐵)

Proof of Theorem onsucssi
StepHypRef Expression
1 onsucssi.1 . 2 𝐴 ∈ On
2 onsucssi.2 . . 3 𝐵 ∈ On
32onordi 4183 . 2 Ord 𝐵
4 ordelsuc 4251 . 2 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))
51, 3, 4mp2an 417 1 (𝐴𝐵 ↔ suc 𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 103   ∈ wcel 1434   ⊆ wss 2974  Ord word 4119  Oncon0 4120  suc csuc 4122 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3406  df-uni 3604  df-tr 3878  df-iord 4123  df-on 4125  df-suc 4128 This theorem is referenced by: (None)
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