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Theorem po0 4039
Description: Any relation is a partial ordering of the empty set. (Contributed by NM, 28-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
po0 𝑅 Po ∅

Proof of Theorem po0
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ral0 3316 . 2 x y z ∅ (¬ x𝑅x ((x𝑅y y𝑅z) → x𝑅z))
2 df-po 4024 . 2 (𝑅 Po ∅ ↔ x y z ∅ (¬ x𝑅x ((x𝑅y y𝑅z) → x𝑅z)))
31, 2mpbir 134 1 𝑅 Po ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wral 2300  c0 3218   class class class wbr 3755   Po wpo 4022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-nul 3219  df-po 4024
This theorem is referenced by:  so0  4054
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