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Theorem ordelon 4275
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 4273 . 2 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
2 elong 4265 . . 3 (𝐵𝐴 → (𝐵 ∈ On ↔ Ord 𝐵))
32adantl 275 . 2 ((Ord 𝐴𝐵𝐴) → (𝐵 ∈ On ↔ Ord 𝐵))
41, 3mpbird 166 1 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 1465  Ord word 4254  Oncon0 4255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260
This theorem is referenced by:  onelon  4276  ordsson  4378  ordpwsucss  4452  tfr1onlemsucfn  6205  tfr1onlemsucaccv  6206  tfr1onlembfn  6209  tfr1onlemubacc  6211  tfr1onlemaccex  6213  tfrcllemsucfn  6218  tfrcllemsucaccv  6219  tfrcllembfn  6222  tfrcllemubacc  6224  tfrcllemaccex  6226  tfrcl  6229
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