MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psref Structured version   Visualization version   GIF version

Theorem psref 17202
Description: A poset is reflexive. (Contributed by NM, 13-May-2008.)
Hypothesis
Ref Expression
psref.1 𝑋 = dom 𝑅
Assertion
Ref Expression
psref ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)

Proof of Theorem psref
StepHypRef Expression
1 psref.1 . . . . 5 𝑋 = dom 𝑅
2 psdmrn 17201 . . . . . 6 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
32simpld 475 . . . . 5 (𝑅 ∈ PosetRel → dom 𝑅 = 𝑅)
41, 3syl5eq 2667 . . . 4 (𝑅 ∈ PosetRel → 𝑋 = 𝑅)
54eleq2d 2686 . . 3 (𝑅 ∈ PosetRel → (𝐴𝑋𝐴 𝑅))
6 pslem 17200 . . . 4 (𝑅 ∈ PosetRel → (((𝐴𝑅𝐴𝐴𝑅𝐴) → 𝐴𝑅𝐴) ∧ (𝐴 𝑅𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐴𝐴𝑅𝐴) → 𝐴 = 𝐴)))
76simp2d 1073 . . 3 (𝑅 ∈ PosetRel → (𝐴 𝑅𝐴𝑅𝐴))
85, 7sylbid 230 . 2 (𝑅 ∈ PosetRel → (𝐴𝑋𝐴𝑅𝐴))
98imp 445 1 ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989   cuni 4434   class class class wbr 4651  dom cdm 5112  ran crn 5113  PosetRelcps 17192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ps 17194
This theorem is referenced by:  psss  17208  psssdm2  17209  ordtt1  21177
  Copyright terms: Public domain W3C validator