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Theorem ordelsuc 4177
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 ((A 𝐶 Ord B) → (A B ↔ suc AB))

Proof of Theorem ordelsuc
StepHypRef Expression
1 ordsucss 4176 . . 3 (Ord B → (A B → suc AB))
21adantl 262 . 2 ((A 𝐶 Ord B) → (A B → suc AB))
3 sucssel 4107 . . 3 (A 𝐶 → (suc ABA B))
43adantr 261 . 2 ((A 𝐶 Ord B) → (suc ABA B))
52, 4impbid 120 1 ((A 𝐶 Ord B) → (A B ↔ suc AB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wcel 1370  wss 2890  Ord word 4044  suc csuc 4047
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-sn 3352  df-uni 3551  df-tr 3825  df-iord 4048  df-suc 4053
This theorem is referenced by:  onsucmin  4178  onsucelsucr  4179  onsucsssucr  4180  onsucsssucexmid  4192  ordgt0ge1  5929  nnsucsssuc  5982
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