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Theorem elon2 4139
 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 4138 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 elex 2611 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
31, 2jca 300 . 2 (𝐴 ∈ On → (Ord 𝐴𝐴 ∈ V))
4 elong 4136 . . 3 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
54biimparc 293 . 2 ((Ord 𝐴𝐴 ∈ V) → 𝐴 ∈ On)
63, 5impbii 124 1 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
 Colors of variables: wff set class Syntax hints:   ∧ wa 102   ↔ wb 103   ∈ wcel 1434  Vcvv 2602  Ord word 4125  Oncon0 4126 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-in 2980  df-ss 2987  df-uni 3610  df-tr 3884  df-iord 4129  df-on 4131 This theorem is referenced by:  tfrexlem  5983
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