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Theorem elon2 4394
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 4393 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 elex 2763 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
31, 2jca 306 . 2 (𝐴 ∈ On → (Ord 𝐴𝐴 ∈ V))
4 elong 4391 . . 3 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
54biimparc 299 . 2 ((Ord 𝐴𝐴 ∈ V) → 𝐴 ∈ On)
63, 5impbii 126 1 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wcel 2160  Vcvv 2752  Ord word 4380  Oncon0 4381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-uni 3825  df-tr 4117  df-iord 4384  df-on 4386
This theorem is referenced by:  tfrexlem  6358  pw1on  7254
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