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Theorem ordelon 4219
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 4217 . 2 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
2 elong 4209 . . 3 (𝐵𝐴 → (𝐵 ∈ On ↔ Ord 𝐵))
32adantl 272 . 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 1439  Ord word 4198  Oncon0 4199
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-in 3006  df-ss 3013  df-uni 3660  df-tr 3943  df-iord 4202  df-on 4204
This theorem is referenced by:  onelon  4220  ordsson  4322  ordpwsucss  4396  tfr1onlemsucfn  6119  tfr1onlemsucaccv  6120  tfr1onlembfn  6123  tfr1onlemubacc  6125  tfr1onlemaccex  6127  tfrcllemsucfn  6132  tfrcllemsucaccv  6133  tfrcllembfn  6136  tfrcllemubacc  6138  tfrcllemaccex  6140  tfrcl  6143
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