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Theorem 0pos 14104
Description: Technical lemma to simplify the statement of ipopos 14279. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 13200) 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 4166 . 2  |-  (/)  e.  _V
2 ral0 3571 . 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 13201 . . 3  |-  (/)  =  (
Base `  (/) )
4 df-ple 13244 . . . 4  |-  le  = Slot  10
54str0 13200 . . 3  |-  (/)  =  ( le `  (/) )
63, 5ispos 14097 . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   (/)c0 3468   class class class wbr 4039   10c10 9819   lecple 13231   Posetcpo 14090
This theorem is referenced by:  ipopos  14279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-slot 13168  df-base 13169  df-ple 13244  df-poset 14096
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