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Theorem ordelsuc 4359
Description: A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.)
Assertion
Ref Expression
ordelsuc ((𝐴𝐶 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))

Proof of Theorem ordelsuc
StepHypRef Expression
1 ordsucss 4358 . . 3 (Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
21adantl 273 . 2 ((𝐴𝐶 ∧ Ord 𝐵) → (𝐴𝐵 → suc 𝐴𝐵))
3 sucssel 4284 . . 3 (𝐴𝐶 → (suc 𝐴𝐵𝐴𝐵))
43adantr 272 . 2 ((𝐴𝐶 ∧ Ord 𝐵) → (suc 𝐴𝐵𝐴𝐵))
52, 4impbid 128 1 ((𝐴𝐶 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 1448  wss 3021  Ord word 4222  suc csuc 4225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480  df-uni 3684  df-tr 3967  df-iord 4226  df-suc 4231
This theorem is referenced by:  onsucssi  4360  onsucmin  4361  onsucelsucr  4362  onsucsssucr  4363  onsucsssucexmid  4380  frecsuclem  6233  ordgt0ge1  6262  nnsucsssuc  6318  ennnfonelemk  11705  nninfsellemeq  12794
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