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Theorem 0pos 14413
Description: Technical lemma to simplify the statement of ipopos 14588. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 13507) 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 4341 . 2  |-  (/)  e.  _V
2 ral0 3734 . 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 13508 . . 3  |-  (/)  =  (
Base `  (/) )
4 df-ple 13551 . . . 4  |-  le  = Slot  10
54str0 13507 . . 3  |-  (/)  =  ( le `  (/) )
63, 5ispos 14406 . 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 888 1  |-  (/)  e.  Poset
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    e. wcel 1726   A.wral 2707   _Vcvv 2958   (/)c0 3630   class class class wbr 4214   10c10 10059   lecple 13538   Posetcpo 14399
This theorem is referenced by:  ipopos  14588
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-slot 13475  df-base 13476  df-ple 13551  df-poset 14405
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