Theorem elong 5958

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 5957	. 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴))
2		df-on 5954	. 2 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	1, 2	elab2g 3559	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))

Syntax hints:   → wi 4   ↔ wb 197   ∈ wcel 2157  Ord word 5949  Oncon0 5950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-v 3404  df-in 3787  df-ss 3794  df-uni 4642  df-tr 4958  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-ord 5953  df-on 5954

This theorem is referenced by:  elon  5959  eloni  5960  elon2  5961  ordelon  5974  onin  5981  limelon  6014  ordsssuc2  6039  onprc  7224  ssonuni  7226  suceloni  7253  ordsuc  7254  oion  8690  hartogs  8698  card2on  8708  tskwe  9069  onssnum  9156  hsmexlem1  9543  ondomon  9680  1stcrestlem  21490  nosupno  32192  hfninf  32636