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Theorem ordelon 4305
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 4303 . 2 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
2 elong 4295 . . 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 1480  Ord word 4284  Oncon0 4285
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290
This theorem is referenced by:  onelon  4306  ordsson  4408  ordpwsucss  4482  tfr1onlemsucfn  6237  tfr1onlemsucaccv  6238  tfr1onlembfn  6241  tfr1onlemubacc  6243  tfr1onlemaccex  6245  tfrcllemsucfn  6250  tfrcllemsucaccv  6251  tfrcllembfn  6254  tfrcllemubacc  6256  tfrcllemaccex  6258  tfrcl  6261
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