Theorem posref 16951

Description: A poset ordering is reflexive. (Contributed by NM, 11-Sep-2011.) (Proof shortened by OpenAI, 25-Mar-2020.)

Hypotheses

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

Assertion

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

Proof of Theorem posref

Step	Hyp	Ref	Expression
1		posprs 16949	. 2 |- ( K e. Poset -> K e. Preset )
2		posi.b	. . 3 |- B = ( Base ` K )
3		posi.l	. . 3 |- .<_ = ( le ` K )
4	2, 3	prsref 16932	. 2 |- ( ( K e. Preset /\ X e. B ) -> X .<_ X )
5	1, 4	sylan 488	1 |- ( ( K e. Poset /\ X e. B ) -> X .<_ X )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   Preset cpreset 16926   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-preset 16928  df-poset 16946

This theorem is referenced by:  posasymb  16952  pleval2  16965  pltval3  16967  pospo  16973  lublecllem  16988  latref  17053  odupos  17135  omndmul2  29712  omndmul  29714  archirngz  29743  gsumle  29779  cvrnbtwn2  34562  cvrnbtwn3  34563  cvrnbtwn4  34566  cvrcmp  34570  llncmp  34808  lplncmp  34848  lvolcmp  34903