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Theorem ordelon 4148
 Description: An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
Assertion
Ref Expression
ordelon ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)

Proof of Theorem ordelon
StepHypRef Expression
1 ordelord 4146 . 2 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
2 elong 4138 . . 3 (𝐵𝐴 → (𝐵 ∈ On ↔ Ord 𝐵))
32adantl 266 . 2 ((Ord 𝐴𝐵𝐴) → (𝐵 ∈ On ↔ Ord 𝐵))
41, 3mpbird 160 1 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   ∈ wcel 1409  Ord word 4127  Oncon0 4128 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133 This theorem is referenced by:  onelon  4149  ordsson  4246  ordpwsucss  4319
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