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Theorem elon2 4079
 Description: An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
Assertion
Ref Expression
elon2 (A On ↔ (Ord A A V))

Proof of Theorem elon2
StepHypRef Expression
1 eloni 4078 . . 3 (A On → Ord A)
2 elex 2560 . . 3 (A On → A V)
31, 2jca 290 . 2 (A On → (Ord A A V))
4 elong 4076 . . 3 (A V → (A On ↔ Ord A))
54biimparc 283 . 2 ((Ord A A V) → A On)
63, 5impbii 117 1 (A On ↔ (Ord A A V))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   ∈ wcel 1390  Vcvv 2551  Ord word 4065  Oncon0 4066 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071 This theorem is referenced by:  tfrexlem  5889
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