Theorem posrefOLD 16275

Description: A poset ordering is reflexive. (Contributed by NM, 11-Sep-2011.) Obsolete version of posref 16274 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
posi.b	|- B = ( Base ` K )
posi.l	|- .<_ = ( le ` K )

Assertion

Ref	Expression
posrefOLD	|- ( ( K e. Poset /\ X e. B ) -> X .<_ X )

Proof of Theorem posrefOLD

Step	Hyp	Ref	Expression
1		id 22	. . . 4 |- ( X e. B -> X e. B )
2	1, 1, 1	3jca 1210	. . 3 |- ( X e. B -> ( X e. B /\ X e. B /\ X e. B ) )
3		posi.b	. . . 4 |- B = ( Base ` K )
4		posi.l	. . . 4 |- .<_ = ( le ` K )
5	3, 4	posi 16273	. . 3 |- ( ( K e. Poset /\ ( X e. B /\ X e. B /\ X e. B ) ) -> ( X .<_ X /\ ( ( X .<_ X /\ X .<_ X ) -> X = X ) /\ ( ( X .<_ X /\ X .<_ X ) -> X .<_ X ) ) )
6	2, 5	sylan2 482	. 2 |- ( ( K e. Poset /\ X e. B ) -> ( X .<_ X /\ ( ( X .<_ X /\ X .<_ X ) -> X = X ) /\ ( ( X .<_ X /\ X .<_ X ) -> X .<_ X ) ) )
7	6	simp1d 1042	1 |- ( ( K e. Poset /\ X e. B ) -> X .<_ X )

Syntax hints:   -> wi 4   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   class class class wbr 4395   ` cfv 5589   Basecbs 15199   lecple 15275   Posetcpo 16263

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-poset 16269

This theorem is referenced by: (None)