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Theorem ordelon 4069
 Description: An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
Assertion
Ref Expression
ordelon ((Ord A B A) → B On)

Proof of Theorem ordelon
StepHypRef Expression
1 ordelord 4067 . 2 ((Ord A B A) → Ord B)
2 elong 4059 . . 3 (B A → (B On ↔ Ord B))
32adantl 262 . 2 ((Ord A B A) → (B On ↔ Ord B))
41, 3mpbird 156 1 ((Ord A B A) → B On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1374  Ord word 4048  Oncon0 4049 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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-in 2901  df-ss 2908  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054 This theorem is referenced by:  onelon  4070  ordsson  4168  ordpwsucss  4227
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