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Theorem ord0 4353
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 4075 . 2 Tr ∅
2 ral0 3496 . 2 𝑥 ∈ ∅ Tr 𝑥
3 dford3 4329 . 2 (Ord ∅ ↔ (Tr ∅ ∧ ∀𝑥 ∈ ∅ Tr 𝑥))
41, 2, 3mpbir2an 927 1 Ord ∅
Colors of variables: wff set class
Syntax hints:  wral 2435  c0 3395  Tr wtr 4064  Ord word 4324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-uni 3775  df-tr 4065  df-iord 4328
This theorem is referenced by:  0elon  4354  ordtriexmidlem  4480  2ordpr  4485  smo0  6247
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