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Theorem elon 4359
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 4358 . 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 2141  Vcvv 2730  Ord word 4347  Oncon0 4348
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353
This theorem is referenced by:  tron  4367  0elon  4377  ordtriexmidlem  4503  ontr2exmid  4509  ordtri2or2exmidlem  4510  onsucelsucexmidlem  4513  exmidonfinlem  7170  bj-omelon  13996
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