Theorem elong 4518

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.

Step	Hyp	Ref	Expression
1		ordeq 4517	. 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2		df-on 4513	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	1, 2	elab2g 2973	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   ∈ wcel 2209  Ord word 4507  Oncon0 4508

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-uni 3936  df-tr 4230  df-iord 4511  df-on 4513

This theorem is used by:  elon  4519  eloni  4520  elon2  4521  ordelon  4528  onin  4531  limelon  4544  ssonuni  4635  onsuc  4648  onsucb  4650  onintonm  4664  onprc  4699  omelon2  4755  bj-nnelon  16985