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Theorem onelon 4266
Description: An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.)
Assertion
Ref Expression
onelon ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)

Proof of Theorem onelon
StepHypRef Expression
1 eloni 4257 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordelon 4265 . 2 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
31, 2sylan 279 1 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1463  Ord word 4244  Oncon0 4245
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-in 3043  df-ss 3050  df-uni 3703  df-tr 3987  df-iord 4248  df-on 4250
This theorem is referenced by:  oneli  4310  ssorduni  4363  unon  4387  tfrlemibacc  6177  tfrlemibxssdm  6178  tfrlemibfn  6179  tfrexlem  6185  tfr1onlemsucaccv  6192  tfrcllemsucaccv  6205  sucinc2  6296  oav2  6313  omv2  6315
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