Theorem ord0 5694

Description: The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)

Assertion

Ref	Expression
ord0	⊢ Ord ∅

Proof of Theorem ord0

Step	Hyp	Ref	Expression
1		tr0 4692	. 2 ⊢ Tr ∅
2		we0 5033	. 2 ⊢ E We ∅
3		df-ord 5643	. 2 ⊢ (Ord ∅ ↔ (Tr ∅ ∧ E We ∅))
4	1, 2, 3	mpbir2an 957	1 ⊢ Ord ∅

Syntax hints:  ∅c0 3874  Tr wtr 4680   E cep 4947   We wwe 4996  Ord word 5639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-uni 4373  df-tr 4681  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643

This theorem is referenced by:  0elon  5695  ord0eln0  5696  ordzsl  6937  smo0  7342  oicl  8317  alephgeom  8788