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Theorem ord0 4376
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 4098 . 2 Tr ∅
2 ral0 3516 . 2 𝑥 ∈ ∅ Tr 𝑥
3 dford3 4352 . 2 (Ord ∅ ↔ (Tr ∅ ∧ ∀𝑥 ∈ ∅ Tr 𝑥))
41, 2, 3mpbir2an 937 1 Ord ∅
Colors of variables: wff set class
Syntax hints:  wral 2448  c0 3414  Tr wtr 4087  Ord word 4347
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-uni 3797  df-tr 4088  df-iord 4351
This theorem is referenced by:  0elon  4377  ordtriexmidlem  4503  2ordpr  4508  smo0  6277
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