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Theorem elong 4076
Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
Assertion
Ref Expression
elong (A 𝑉 → (A On ↔ Ord A))

Proof of Theorem elong
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 ordeq 4075 . 2 (x = A → (Ord x ↔ Ord A))
2 df-on 4071 . 2 On = {x ∣ Ord x}
31, 2elab2g 2683 1 (A 𝑉 → (A On ↔ Ord A))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wcel 1390  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:  elon  4077  eloni  4078  elon2  4079  ordelon  4086  onin  4089  limelon  4102  ssonuni  4180  suceloni  4193  sucelon  4195  onprc  4230  omelon2  4273
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