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Theorem elon2 4113
Description: An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
Assertion
Ref Expression
elon2 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))

Proof of Theorem elon2
StepHypRef Expression
1 eloni 4112 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 elex 2566 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
31, 2jca 290 . 2 (𝐴 ∈ On → (Ord 𝐴𝐴 ∈ V))
4 elong 4110 . . 3 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
54biimparc 283 . 2 ((Ord 𝐴𝐴 ∈ V) → 𝐴 ∈ On)
63, 5impbii 117 1 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
Colors of variables: wff set class
Syntax hints:  wa 97  wb 98  wcel 1393  Vcvv 2557  Ord word 4099  Oncon0 4100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105
This theorem is referenced by:  tfrexlem  5948
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