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Theorem elon 4266
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 4265 . 2 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
31, 2ax-mp 5 1 (𝐴 ∈ On ↔ Ord 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 1465  Vcvv 2660  Ord word 4254  Oncon0 4255
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260
This theorem is referenced by:  tron  4274  0elon  4284  ordtriexmidlem  4405  ontr2exmid  4410  ordtri2or2exmidlem  4411  onsucelsucexmidlem  4414  exmidonfinlem  7017  bj-omelon  13086
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