Theorem elong 6199

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 6198	. 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2		df-on 6195	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	1, 2	elab2g 3668	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2114  Ord word 6190  Oncon0 6191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3496  df-in 3943  df-ss 3952  df-uni 4839  df-tr 5173  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-ord 6194  df-on 6195

This theorem is referenced by:  elon  6200  eloni  6201  elon2  6202  ordelon  6215  onin  6222  limelon  6254  ordsssuc2  6279  onprc  7499  ssonuni  7501  suceloni  7528  ordsuc  7529  oion  9000  hartogs  9008  card2on  9018  tskwe  9379  onssnum  9466  hsmexlem1  9848  ondomon  9985  1stcrestlem  22060  nosupno  33203  hfninf  33647