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Theorem elon2 4359
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 4358 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 elex 2741 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
31, 2jca 304 . 2 (𝐴 ∈ On → (Ord 𝐴𝐴 ∈ V))
4 elong 4356 . . 3 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
54biimparc 297 . 2 ((Ord 𝐴𝐴 ∈ V) → 𝐴 ∈ On)
63, 5impbii 125 1 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wcel 2141  Vcvv 2730  Ord word 4345  Oncon0 4346
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 3795  df-tr 4086  df-iord 4349  df-on 4351
This theorem is referenced by:  tfrexlem  6310  pw1on  7190
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