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Theorem elon 4111
 Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
Hypothesis
Ref Expression
elon.1 𝐴 ∈ V
Assertion
Ref Expression
elon (𝐴 ∈ On ↔ Ord 𝐴)

Proof of Theorem elon
StepHypRef Expression
1 elon.1 . 2 𝐴 ∈ V
2 elong 4110 . 2 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
31, 2ax-mp 7 1 (𝐴 ∈ On ↔ Ord 𝐴)
 Colors of variables: wff set class Syntax hints:   ↔ 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:  tron  4119  0elon  4129  ordtriexmidlem  4245  ontr2exmid  4250  ordtri2or2exmidlem  4251  onsucelsucexmidlem  4254  bj-omelon  10086
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