Theorem 0pos 16954

Description: Technical lemma to simplify the statement of ipopos 17160. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 15911) 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.

Step	Hyp	Ref	Expression
1		0ex 4790	. 2 |- (/) e. _V
2		ral0 4076	. 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 15912	. . 3 |- (/) = ( Base ` (/) )
4		df-ple 15961	. . . 4 |- le = Slot ; 1 0
5	4	str0 15911	. . 3 |- (/) = ( le ` (/) )
6	3, 5	ispos 16947	. 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 ) ) ) )
7	1, 2, 6	mpbir2an 955	1 |- (/) e. Poset

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   A.wral 2912   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   0cc0 9936   1c1 9937  ;cdc 11493   lecple 15948   Posetcpo 16940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-slot 15861  df-base 15863  df-ple 15961  df-poset 16946

This theorem is referenced by:  ipopos  17160