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Theorem elong 4372
Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
Assertion
Ref Expression
elong (𝐴𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Proof of Theorem elong
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ordeq 4371 . 2 (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2 df-on 4367 . 2 On = {𝑥 ∣ Ord 𝑥}
31, 2elab2g 2884 1 (𝐴𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wcel 2148  Ord word 4361  Oncon0 4362
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-in 3135  df-ss 3142  df-uni 3810  df-tr 4101  df-iord 4365  df-on 4367
This theorem is referenced by:  elon  4373  eloni  4374  elon2  4375  ordelon  4382  onin  4385  limelon  4398  ssonuni  4486  onsuc  4499  onsucb  4501  onintonm  4515  onprc  4550  omelon2  4606  bj-nnelon  14562
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