ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elon2 GIF version

Theorem elon2 4227
Description: An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
Assertion
Ref Expression
elon2 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))

Proof of Theorem elon2
StepHypRef Expression
1 eloni 4226 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 elex 2644 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
31, 2jca 301 . 2 (𝐴 ∈ On → (Ord 𝐴𝐴 ∈ V))
4 elong 4224 . . 3 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
54biimparc 294 . 2 ((Ord 𝐴𝐴 ∈ V) → 𝐴 ∈ On)
63, 5impbii 125 1 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wcel 1445  Vcvv 2633  Ord word 4213  Oncon0 4214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-in 3019  df-ss 3026  df-uni 3676  df-tr 3959  df-iord 4217  df-on 4219
This theorem is referenced by:  tfrexlem  6137
  Copyright terms: Public domain W3C validator