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Theorem ord0 6265
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 5172 . 2 Tr ∅
2 we0 5546 . 2 E We ∅
3 df-ord 6216 . 2 (Ord ∅ ↔ (Tr ∅ ∧ E We ∅))
41, 2, 3mpbir2an 711 1 Ord ∅
Colors of variables: wff setvar class
Syntax hints:  c0 4237  Tr wtr 5161   E cep 5459   We wwe 5508  Ord word 6212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-tr 5162  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-ord 6216
This theorem is referenced by:  0elon  6266  ord0eln0  6267  ordzsl  7624  smo0  8095  oicl  9145  alephgeom  9696
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