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Theorem ord0 5741
 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 4729 . 2 Tr ∅
2 we0 5074 . 2 E We ∅
3 df-ord 5690 . 2 (Ord ∅ ↔ (Tr ∅ ∧ E We ∅))
41, 2, 3mpbir2an 954 1 Ord ∅
 Colors of variables: wff setvar class Syntax hints:  ∅c0 3896  Tr wtr 4717   E cep 4988   We wwe 5037  Ord word 5686 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-in 3566  df-ss 3573  df-nul 3897  df-pw 4137  df-uni 4408  df-tr 4718  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690 This theorem is referenced by:  0elon  5742  ord0eln0  5743  ordzsl  6999  smo0  7407  oicl  8386  alephgeom  8857
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