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Theorem 0pos 14088
Description: Technical lemma to simplify the statement of ipopos 14263. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 13184) and any relation partially orders an empty set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
0pos  |-  (/)  e.  Poset

Proof of Theorem 0pos
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4150 . 2  |-  (/)  e.  _V
2 ral0 3558 . 2  |-  A. a  e.  (/)  A. b  e.  (/)  A. c  e.  (/)  ( a (/) a  /\  ( ( a (/) b  /\  b (/) a )  ->  a  =  b )  /\  ( ( a (/) b  /\  b (/) c )  ->  a (/) c ) )
3 base0 13185 . . 3  |-  (/)  =  (
Base `  (/) )
4 df-ple 13228 . . . 4  |-  le  = Slot  10
54str0 13184 . . 3  |-  (/)  =  ( le `  (/) )
63, 5ispos 14081 . 2  |-  ( (/)  e.  Poset 
<->  ( (/)  e.  _V  /\ 
A. a  e.  (/)  A. b  e.  (/)  A. c  e.  (/)  ( a (/) a  /\  ( ( a
(/) b  /\  b (/) a )  ->  a  =  b )  /\  ( ( a (/) b  /\  b (/) c )  ->  a (/) c ) ) ) )
71, 2, 6mpbir2an 886 1  |-  (/)  e.  Poset
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   (/)c0 3455   class class class wbr 4023   10c10 9803   lecple 13215   Posetcpo 14074
This theorem is referenced by:  ipopos  14263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-slot 13152  df-base 13153  df-ple 13228  df-poset 14080
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