Theorem isposi 16956

Description: Properties that determine a poset (implicit structure version). (Contributed by NM, 11-Sep-2011.)

Hypotheses

Ref	Expression
isposi.k	|- K e. _V
isposi.b	|- B = ( Base ` K )
isposi.l	|- .<_ = ( le ` K )
isposi.1	|- ( x e. B -> x .<_ x )
isposi.2	|- ( ( x e. B /\ y e. B ) -> ( ( x .<_ y /\ y .<_ x ) -> x = y ) )
isposi.3	|- ( ( x e. B /\ y e. B /\ z e. B ) -> ( ( x .<_ y /\ y .<_ z ) -> x .<_ z ) )

Assertion

Ref	Expression
isposi	|- K e. Poset

Distinct variable groups:   x, y, z, B   x, .<_ , y, z

Allowed substitution hints:   K( x, y, z)

Proof of Theorem isposi

Step	Hyp	Ref	Expression
1		isposi.k	. 2 |- K e. _V
2		isposi.1	. . . . 5 |- ( x e. B -> x .<_ x )
3	2	3ad2ant1 1082	. . . 4 |- ( ( x e. B /\ y e. B /\ z e. B ) -> x .<_ x )
4		isposi.2	. . . . 5 |- ( ( x e. B /\ y e. B ) -> ( ( x .<_ y /\ y .<_ x ) -> x = y ) )
5	4	3adant3 1081	. . . 4 |- ( ( x e. B /\ y e. B /\ z e. B ) -> ( ( x .<_ y /\ y .<_ x ) -> x = y ) )
6		isposi.3	. . . 4 |- ( ( x e. B /\ y e. B /\ z e. B ) -> ( ( x .<_ y /\ y .<_ z ) -> x .<_ z ) )
7	3, 5, 6	3jca 1242	. . 3 |- ( ( x e. B /\ y e. B /\ z e. B ) -> ( x .<_ x /\ ( ( x .<_ y /\ y .<_ x ) -> x = y ) /\ ( ( x .<_ y /\ y .<_ z ) -> x .<_ z ) ) )
8	7	rgen3 2976	. 2 |- A. x e. B A. y e. B A. z e. B ( x .<_ x /\ ( ( x .<_ y /\ y .<_ x ) -> x = y ) /\ ( ( x .<_ y /\ y .<_ z ) -> x .<_ z ) )
9		isposi.b	. . 3 |- B = ( Base ` K )
10		isposi.l	. . 3 |- .<_ = ( le ` K )
11	9, 10	ispos 16947	. 2 |- ( K e. Poset <-> ( K e. _V /\ A. x e. B A. y e. B A. z e. B ( x .<_ x /\ ( ( x .<_ y /\ y .<_ x ) -> x = y ) /\ ( ( x .<_ y /\ y .<_ z ) -> x .<_ z ) ) ) )
12	1, 8, 11	mpbir2an 955	1 |- K e. Poset

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   class class class wbr 4653   ` cfv 5888   Basecbs 15857   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

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-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-iota 5851  df-fv 5896  df-poset 16946

This theorem is referenced by:  isposix  16957  xrstos  29679  xrge0omnd  29711