MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ord0 Structured version   Unicode version

Theorem ord0 4861
Description: The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)
Assertion
Ref Expression
ord0  |-  Ord  (/)

Proof of Theorem ord0
StepHypRef Expression
1 tr0 4488 . 2  |-  Tr  (/)
2 we0 4805 . 2  |-  _E  We  (/)
3 df-ord 4812 . 2  |-  ( Ord  (/) 
<->  ( Tr  (/)  /\  _E  We  (/) ) )
41, 2, 3mpbir2an 918 1  |-  Ord  (/)
Colors of variables: wff setvar class
Syntax hints:   (/)c0 3728   Tr wtr 4477    _E cep 4720    We wwe 4768   Ord word 4808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-dif 3409  df-in 3413  df-ss 3420  df-nul 3729  df-pw 3946  df-uni 4181  df-tr 4478  df-po 4731  df-so 4732  df-fr 4769  df-we 4771  df-ord 4812
This theorem is referenced by:  0elon  4862  ord0eln0  4863  ordzsl  6601  smo0  6969  oicl  7891  alephgeom  8398
  Copyright terms: Public domain W3C validator