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Theorem ord0 4156
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 3893 . 2 Tr ∅
2 ral0 3350 . 2 𝑥 ∈ ∅ Tr 𝑥
3 dford3 4132 . 2 (Ord ∅ ↔ (Tr ∅ ∧ ∀𝑥 ∈ ∅ Tr 𝑥))
41, 2, 3mpbir2an 860 1 Ord ∅
Colors of variables: wff set class
Syntax hints:  wral 2323  c0 3252  Tr wtr 3882  Ord word 4127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-dif 2948  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-uni 3609  df-tr 3883  df-iord 4131
This theorem is referenced by:  0elon  4157  ordtriexmidlem  4273  2ordpr  4277  smo0  5944
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